Solved on Jan 22, 2024

Estimate and solve for x in equations: a) $35=5(3^{x})$ , b) $117=3\times2^{x}$ , c) $1000=500(1+0.08)^{x}$ , d) $45=100(1/2)^{x}$ . Round answers to 3 decimal places.

STEP 1

Assumptions
1. We will solve each equation for $x$ .
2. We will use logarithms to solve the exponential equations.
3. We will round the final answers to three decimal places.

STEP 2

Start with equation (a) $35=5\left(3^{x}\right)$ .

STEP 3

Divide both sides of the equation by 5 to isolate the exponential term.
$3^{x} = \frac{35}{5}$

STEP 4

Calculate the right side of the equation.
$3^{x} = 7$

STEP 5

Take the natural logarithm of both sides to solve for $x$ .
$\ln(3^{x}) = \ln(7)$

STEP 6

Use the logarithmic property $\ln(a^{b}) = b \cdot \ln(a)$ to simplify the left side of the equation.
$x \cdot \ln(3) = \ln(7)$

STEP 7

Divide both sides by $\ln(3)$ to solve for $x$ .
$x = \frac{\ln(7)}{\ln(3)}$

STEP 8

Calculate the value of $x$ using a calculator and round to three decimal places.
$x \approx \frac{\ln(7)}{\ln(3)} \approx 1.771$

STEP 9

Now solve equation (c) $1000=500(1+0.08)^{x}$ .

STEP 10

Divide both sides of the equation by 500 to isolate the exponential term.
$(1+0.08)^{x} = \frac{1000}{500}$

STEP 11

Calculate the right side of the equation.
$(1+0.08)^{x} = 2$

STEP 12

Take the natural logarithm of both sides to solve for $x$ .
$\ln((1+0.08)^{x}) = \ln(2)$

STEP 13

Use the logarithmic property $\ln(a^{b}) = b \cdot \ln(a)$ to simplify the left side of the equation.
$x \cdot \ln(1.08) = \ln(2)$

STEP 14

Divide both sides by $\ln(1.08)$ to solve for $x$ .
$x = \frac{\ln(2)}{\ln(1.08)}$

STEP 15

Calculate the value of $x$ using a calculator and round to three decimal places.
$x \approx \frac{\ln(2)}{\ln(1.08)} \approx 9.006$

STEP 16

Now solve equation (b) $117=3 \times 2^{x}$ .

STEP 17

Divide both sides of the equation by 3 to isolate the exponential term.
$2^{x} = \frac{117}{3}$

STEP 18

Calculate the right side of the equation.
$2^{x} = 39$

STEP 19

Take the natural logarithm of both sides to solve for $x$ .
$\ln(2^{x}) = \ln(39)$

STEP 20

Use the logarithmic property $\ln(a^{b}) = b \cdot \ln(a)$ to simplify the left side of the equation.
$x \cdot \ln(2) = \ln(39)$

STEP 21

Divide both sides by $\ln(2)$ to solve for $x$ .
$x = \frac{\ln(39)}{\ln(2)}$

STEP 22

Calculate the value of $x$ using a calculator and round to three decimal places.
$x \approx \frac{\ln(39)}{\ln(2)} \approx 5.285$

STEP 23

Finally, solve equation (d) $45=100\left(\frac{1}{2}\right)^{x}$ .

STEP 24

Divide both sides of the equation by 100 to isolate the exponential term.
$\left(\frac{1}{2}\right)^{x} = \frac{45}{100}$

STEP 25

Calculate the right side of the equation.
$\left(\frac{1}{2}\right)^{x} = 0.45$

STEP 26

Take the natural logarithm of both sides to solve for $x$ .
$\ln\left(\left(\frac{1}{2}\right)^{x}\right) = \ln(0.45)$

STEP 27

Use the logarithmic property $\ln(a^{b}) = b \cdot \ln(a)$ to simplify the left side of the equation.
$x \cdot \ln\left(\frac{1}{2}\right) = \ln(0.45)$

STEP 28

Divide both sides by $\ln\left(\frac{1}{2}\right)$ to solve for $x$ .
$x = \frac{\ln(0.45)}{\ln\left(\frac{1}{2}\right)}$

STEP 29

Calculate the value of $x$ using a calculator and round to three decimal places.
$x \approx \frac{\ln(0.45)}{\ln\left(\frac{1}{2}\right)} \approx 1.137$
The solutions for each equation are:
a) $x \approx 1.771$ c) $x \approx 9.006$ b) $x \approx 5.285$ d) $x \approx 1.137$

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Estimate and solve for x in equations: a) 35=5(3x)35=5(3^{x})35=5(3x), b) 117=3×2x117=3\times2^{x}117=3×2x, c) 1000=500(1+0.08)x1000=500(1+0.08)^{x}1000=500(1+0.08)x, d) 45=100(1/2)x45=100(1/2)^{x}45=100(1/2)x. Round answers to 3 decimal places.

STEP 1

Assumptions1. We will solve each equation for xxx.2. We will use logarithms to solve the exponential equations.3. We will round the final answers to three decimal places.

STEP 2

Start with equation (a) 35=5(3x)35=5\left(3^{x}\right)35=5(3x).

STEP 3

Divide both sides of the equation by 5 to isolate the exponential term.3x=3553^{x} = \frac{35}{5}3x=535​

STEP 4

Calculate the right side of the equation.3x=73^{x} = 73x=7

STEP 5

Take the natural logarithm of both sides to solve for xxx.ln⁡(3x)=ln⁡(7)\ln(3^{x}) = \ln(7)ln(3x)=ln(7)

STEP 6

Use the logarithmic property ln⁡(ab)=b⋅ln⁡(a)\ln(a^{b}) = b \cdot \ln(a)ln(ab)=b⋅ln(a) to simplify the left side of the equation.x⋅ln⁡(3)=ln⁡(7)x \cdot \ln(3) = \ln(7)x⋅ln(3)=ln(7)

STEP 7

Divide both sides by ln⁡(3)\ln(3)ln(3) to solve for xxx.x=ln⁡(7)ln⁡(3)x = \frac{\ln(7)}{\ln(3)}x=ln(3)ln(7)​

STEP 8

Calculate the value of xxx using a calculator and round to three decimal places.x≈ln⁡(7)ln⁡(3)≈1.771x \approx \frac{\ln(7)}{\ln(3)} \approx 1.771x≈ln(3)ln(7)​≈1.771

STEP 9

Now solve equation (c) 1000=500(1+0.08)x1000=500(1+0.08)^{x}1000=500(1+0.08)x.

STEP 10

Divide both sides of the equation by 500 to isolate the exponential term.(1+0.08)x=1000500(1+0.08)^{x} = \frac{1000}{500}(1+0.08)x=5001000​

STEP 11

Calculate the right side of the equation.(1+0.08)x=2(1+0.08)^{x} = 2(1+0.08)x=2

STEP 12

Take the natural logarithm of both sides to solve for xxx.ln⁡((1+0.08)x)=ln⁡(2)\ln((1+0.08)^{x}) = \ln(2)ln((1+0.08)x)=ln(2)

STEP 13

Use the logarithmic property ln⁡(ab)=b⋅ln⁡(a)\ln(a^{b}) = b \cdot \ln(a)ln(ab)=b⋅ln(a) to simplify the left side of the equation.x⋅ln⁡(1.08)=ln⁡(2)x \cdot \ln(1.08) = \ln(2)x⋅ln(1.08)=ln(2)

STEP 14

Divide both sides by ln⁡(1.08)\ln(1.08)ln(1.08) to solve for xxx.x=ln⁡(2)ln⁡(1.08)x = \frac{\ln(2)}{\ln(1.08)}x=ln(1.08)ln(2)​

STEP 15

Calculate the value of xxx using a calculator and round to three decimal places.x≈ln⁡(2)ln⁡(1.08)≈9.006x \approx \frac{\ln(2)}{\ln(1.08)} \approx 9.006x≈ln(1.08)ln(2)​≈9.006

STEP 16

Now solve equation (b) 117=3×2x117=3 \times 2^{x}117=3×2x.

STEP 17

Divide both sides of the equation by 3 to isolate the exponential term.2x=11732^{x} = \frac{117}{3}2x=3117​

STEP 18

Calculate the right side of the equation.2x=392^{x} = 392x=39

STEP 19

Take the natural logarithm of both sides to solve for xxx.ln⁡(2x)=ln⁡(39)\ln(2^{x}) = \ln(39)ln(2x)=ln(39)

STEP 20

Use the logarithmic property ln⁡(ab)=b⋅ln⁡(a)\ln(a^{b}) = b \cdot \ln(a)ln(ab)=b⋅ln(a) to simplify the left side of the equation.x⋅ln⁡(2)=ln⁡(39)x \cdot \ln(2) = \ln(39)x⋅ln(2)=ln(39)

STEP 21

Divide both sides by ln⁡(2)\ln(2)ln(2) to solve for xxx.x=ln⁡(39)ln⁡(2)x = \frac{\ln(39)}{\ln(2)}x=ln(2)ln(39)​

STEP 22

Calculate the value of xxx using a calculator and round to three decimal places.x≈ln⁡(39)ln⁡(2)≈5.285x \approx \frac{\ln(39)}{\ln(2)} \approx 5.285x≈ln(2)ln(39)​≈5.285

STEP 23

Finally, solve equation (d) 45=100(12)x45=100\left(\frac{1}{2}\right)^{x}45=100(21​)x.

STEP 24

Divide both sides of the equation by 100 to isolate the exponential term.(12)x=45100\left(\frac{1}{2}\right)^{x} = \frac{45}{100}(21​)x=10045​

STEP 25

Calculate the right side of the equation.(12)x=0.45\left(\frac{1}{2}\right)^{x} = 0.45(21​)x=0.45

STEP 26

Take the natural logarithm of both sides to solve for xxx.ln⁡((12)x)=ln⁡(0.45)\ln\left(\left(\frac{1}{2}\right)^{x}\right) = \ln(0.45)ln((21​)x)=ln(0.45)

STEP 27

Use the logarithmic property ln⁡(ab)=b⋅ln⁡(a)\ln(a^{b}) = b \cdot \ln(a)ln(ab)=b⋅ln(a) to simplify the left side of the equation.x⋅ln⁡(12)=ln⁡(0.45)x \cdot \ln\left(\frac{1}{2}\right) = \ln(0.45)x⋅ln(21​)=ln(0.45)

STEP 28

Divide both sides by ln⁡(12)\ln\left(\frac{1}{2}\right)ln(21​) to solve for xxx.x=ln⁡(0.45)ln⁡(12)x = \frac{\ln(0.45)}{\ln\left(\frac{1}{2}\right)}x=ln(21​)ln(0.45)​

STEP 29

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Estimate and solve for x in equations: a) $35=5(3^{x})$ , b) $117=3\times2^{x}$ , c) $1000=500(1+0.08)^{x}$ , d) $45=100(1/2)^{x}$ . Round answers to 3 decimal places.

Assumptions
1. We will solve each equation for $x$ .
2. We will use logarithms to solve the exponential equations.
3. We will round the final answers to three decimal places.

Start with equation (a) $35=5\left(3^{x}\right)$ .

Divide both sides of the equation by 5 to isolate the exponential term.
$3^{x} = \frac{35}{5}$

Calculate the right side of the equation.
$3^{x} = 7$

Take the natural logarithm of both sides to solve for $x$ .
$\ln(3^{x}) = \ln(7)$

Use the logarithmic property $\ln(a^{b}) = b \cdot \ln(a)$ to simplify the left side of the equation.
$x \cdot \ln(3) = \ln(7)$

Divide both sides by $\ln(3)$ to solve for $x$ .
$x = \frac{\ln(7)}{\ln(3)}$

Calculate the value of $x$ using a calculator and round to three decimal places.
$x \approx \frac{\ln(7)}{\ln(3)} \approx 1.771$

Now solve equation (c) $1000=500(1+0.08)^{x}$ .

Divide both sides of the equation by 500 to isolate the exponential term.
$(1+0.08)^{x} = \frac{1000}{500}$

Calculate the right side of the equation.
$(1+0.08)^{x} = 2$

Take the natural logarithm of both sides to solve for $x$ .
$\ln((1+0.08)^{x}) = \ln(2)$

Use the logarithmic property $\ln(a^{b}) = b \cdot \ln(a)$ to simplify the left side of the equation.
$x \cdot \ln(1.08) = \ln(2)$

Divide both sides by $\ln(1.08)$ to solve for $x$ .
$x = \frac{\ln(2)}{\ln(1.08)}$

Calculate the value of $x$ using a calculator and round to three decimal places.
$x \approx \frac{\ln(2)}{\ln(1.08)} \approx 9.006$

Now solve equation (b) $117=3 \times 2^{x}$ .

Divide both sides of the equation by 3 to isolate the exponential term.
$2^{x} = \frac{117}{3}$

Calculate the right side of the equation.
$2^{x} = 39$

Take the natural logarithm of both sides to solve for $x$ .
$\ln(2^{x}) = \ln(39)$

Use the logarithmic property $\ln(a^{b}) = b \cdot \ln(a)$ to simplify the left side of the equation.
$x \cdot \ln(2) = \ln(39)$

Divide both sides by $\ln(2)$ to solve for $x$ .
$x = \frac{\ln(39)}{\ln(2)}$

Calculate the value of $x$ using a calculator and round to three decimal places.
$x \approx \frac{\ln(39)}{\ln(2)} \approx 5.285$

Finally, solve equation (d) $45=100\left(\frac{1}{2}\right)^{x}$ .

Divide both sides of the equation by 100 to isolate the exponential term.
$\left(\frac{1}{2}\right)^{x} = \frac{45}{100}$

Calculate the right side of the equation.
$\left(\frac{1}{2}\right)^{x} = 0.45$

Take the natural logarithm of both sides to solve for $x$ .
$\ln\left(\left(\frac{1}{2}\right)^{x}\right) = \ln(0.45)$

Use the logarithmic property $\ln(a^{b}) = b \cdot \ln(a)$ to simplify the left side of the equation.
$x \cdot \ln\left(\frac{1}{2}\right) = \ln(0.45)$

Divide both sides by $\ln\left(\frac{1}{2}\right)$ to solve for $x$ .
$x = \frac{\ln(0.45)}{\ln\left(\frac{1}{2}\right)}$