Math / Algebra

QuestionSolve the exponential equation. $4(1.19)^{x}+2=15$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution is $x=$ $\square$ (Do not round until the final answer. Then round to four decimal places as needed. Use a comma to separate answers as needed.) B. There is no solution.

Studdy Solution

STEP 1

1. The equation $4(1.19)^{x} + 2 = 15$ is an exponential equation.
2. We will need to isolate the exponential term and use logarithms to solve for $x$ .

STEP 2

1. Isolate the exponential term $(1.19)^{x}$ .
2. Use logarithms to solve for $x$ .
3. Calculate the value of $x$ and round to four decimal places.

STEP 3

First, subtract 2 from both sides of the equation to isolate the term with the exponent:
$4(1.19)^{x} + 2 = 15$ $4(1.19)^{x} = 15 - 2$ $4(1.19)^{x} = 13$

STEP 4

Next, divide both sides by 4 to further isolate the exponential term:
$4(1.19)^{x} = 13$ $(1.19)^{x} = \frac{13}{4}$ $(1.19)^{x} = 3.25$

STEP 5

To solve for $x$ , take the logarithm of both sides. You can use the natural logarithm (ln) or common logarithm (log). Here, we'll use the natural logarithm:
$\ln((1.19)^{x}) = \ln(3.25)$
Using the power rule of logarithms, $\ln(a^b) = b \ln(a)$ , we have:
$x \cdot \ln(1.19) = \ln(3.25)$

STEP 6

Solve for $x$ by dividing both sides by $\ln(1.19)$ :
$x = \frac{\ln(3.25)}{\ln(1.19)}$

STEP 7

Calculate the value of $x$ using a calculator, and round to four decimal places:
$x \approx \frac{\ln(3.25)}{\ln(1.19)} \approx 6.7993$
The solution is $x = 6.7993$ .
A. The solution is $x = \boxed{6.7993}$ .

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Studdy Solution

STEP 1

1. The equation $4(1.19)^{x} + 2 = 15$ is an exponential equation.
2. We will need to isolate the exponential term and use logarithms to solve for $x$ .

STEP 2

1. Isolate the exponential term $(1.19)^{x}$ .
2. Use logarithms to solve for $x$ .
3. Calculate the value of $x$ and round to four decimal places.

STEP 3

First, subtract 2 from both sides of the equation to isolate the term with the exponent:
$4(1.19)^{x} + 2 = 15$ $4(1.19)^{x} = 15 - 2$ $4(1.19)^{x} = 13$

STEP 4

Next, divide both sides by 4 to further isolate the exponential term:
$4(1.19)^{x} = 13$ $(1.19)^{x} = \frac{13}{4}$ $(1.19)^{x} = 3.25$

STEP 5

To solve for $x$ , take the logarithm of both sides. You can use the natural logarithm (ln) or common logarithm (log). Here, we'll use the natural logarithm:
$\ln((1.19)^{x}) = \ln(3.25)$
Using the power rule of logarithms, $\ln(a^b) = b \ln(a)$ , we have:
$x \cdot \ln(1.19) = \ln(3.25)$

STEP 6

Solve for $x$ by dividing both sides by $\ln(1.19)$ :
$x = \frac{\ln(3.25)}{\ln(1.19)}$

STEP 7

Calculate the value of $x$ using a calculator, and round to four decimal places:
$x \approx \frac{\ln(3.25)}{\ln(1.19)} \approx 6.7993$
The solution is $x = 6.7993$ .
A. The solution is $x = \boxed{6.7993}$ .

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Studdy Solution

STEP 1

1. The equation 4(1.19)x+2=15 4(1.19)^{x} + 2 = 15 4(1.19)x+2=15 is an exponential equation.2. We will need to isolate the exponential term and use logarithms to solve for x x x.

STEP 2

1. Isolate the exponential term (1.19)x (1.19)^{x} (1.19)x.2. Use logarithms to solve for x x x.3. Calculate the value of x x x and round to four decimal places.

STEP 3

First, subtract 2 from both sides of the equation to isolate the term with the exponent:4(1.19)x+2=15 4(1.19)^{x} + 2 = 15 4(1.19)x+2=15 4(1.19)x=15−2 4(1.19)^{x} = 15 - 2 4(1.19)x=15−2 4(1.19)x=13 4(1.19)^{x} = 13 4(1.19)x=13

STEP 4

Next, divide both sides by 4 to further isolate the exponential term:4(1.19)x=13 4(1.19)^{x} = 13 4(1.19)x=13 (1.19)x=134 (1.19)^{x} = \frac{13}{4} (1.19)x=413​ (1.19)x=3.25 (1.19)^{x} = 3.25 (1.19)x=3.25

STEP 5

STEP 6

Solve for x x x by dividing both sides by ln⁡(1.19) \ln(1.19) ln(1.19):x=ln⁡(3.25)ln⁡(1.19) x = \frac{\ln(3.25)}{\ln(1.19)} x=ln(1.19)ln(3.25)​

STEP 7

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1. The equation $4(1.19)^{x} + 2 = 15$ is an exponential equation.
2. We will need to isolate the exponential term and use logarithms to solve for $x$ .

1. Isolate the exponential term $(1.19)^{x}$ .
2. Use logarithms to solve for $x$ .
3. Calculate the value of $x$ and round to four decimal places.

First, subtract 2 from both sides of the equation to isolate the term with the exponent:
$4(1.19)^{x} + 2 = 15$ $4(1.19)^{x} = 15 - 2$ $4(1.19)^{x} = 13$

Next, divide both sides by 4 to further isolate the exponential term:
$4(1.19)^{x} = 13$ $(1.19)^{x} = \frac{13}{4}$ $(1.19)^{x} = 3.25$

Solve for $x$ by dividing both sides by $\ln(1.19)$ :
$x = \frac{\ln(3.25)}{\ln(1.19)}$