Question Solve for the exact value of $x$ where $\log_7(4x) + 3\log_7(5) = 5$ .

Studdy Solution

STEP 1

Assumptions1. The base of the logarithm is7. . The logarithm of the product of two numbers is equal to the sum of the logarithms of each number.
3. The logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number.

STEP 2

We can use the property of logarithms that states the logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number. This allows us to rewrite the second term on the left side of the equation.
$\log{7}(5) = \log{7}(5^)$

STEP 3

Now, substitute $\log{7}(5^3)$ back into the original equation.
$\log{7}( x)+\log{7}(5^3)=5$

STEP 4

We can use the property of logarithms that states the logarithm of the product of two numbers is equal to the sum of the logarithms of each number. This allows us to combine the two logarithms on the left side of the equation.
$\log{7}(4 x \cdot^3)=$

STEP 5

implify the inside of the logarithm.
$\log{7}(500x)=5$

STEP 6

To remove the logarithm, we can use the property of logarithms that states $b^{\log{b}(x)}=x$ . This gives us $^5 =500x$

STEP 7

To solve for $x$ , we can divide both sides of the equation by500.
$x = \frac{7^5}{500}$

STEP 8

Finally, calculate the value of $x$ .
$x = \frac{7^5}{500} = \frac{16807}{500} =33.614$ So the exact value of $x$ is $33.614$ .

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Question Solve for the exact value of $x$ where $\log_7(4x) + 3\log_7(5) = 5$ .

Studdy Solution

STEP 1

Assumptions1. The base of the logarithm is7. . The logarithm of the product of two numbers is equal to the sum of the logarithms of each number.
3. The logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number.

STEP 2

We can use the property of logarithms that states the logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number. This allows us to rewrite the second term on the left side of the equation.
$\log{7}(5) = \log{7}(5^)$

STEP 3

Now, substitute $\log{7}(5^3)$ back into the original equation.
$\log{7}( x)+\log{7}(5^3)=5$

STEP 4

We can use the property of logarithms that states the logarithm of the product of two numbers is equal to the sum of the logarithms of each number. This allows us to combine the two logarithms on the left side of the equation.
$\log{7}(4 x \cdot^3)=$

STEP 5

implify the inside of the logarithm.
$\log{7}(500x)=5$

STEP 6

To remove the logarithm, we can use the property of logarithms that states $b^{\log{b}(x)}=x$ . This gives us $^5 =500x$

STEP 7

To solve for $x$ , we can divide both sides of the equation by500.
$x = \frac{7^5}{500}$

STEP 8

Finally, calculate the value of $x$ .
$x = \frac{7^5}{500} = \frac{16807}{500} =33.614$ So the exact value of $x$ is $33.614$ .

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Question Solve for the exact value of xxx where log⁡7(4x)+3log⁡7(5)=5\log_7(4x) + 3\log_7(5) = 5log7​(4x)+3log7​(5)=5.

Studdy Solution

STEP 1

Assumptions1. The base of the logarithm is7. . The logarithm of the product of two numbers is equal to the sum of the logarithms of each number.3. The logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number.

STEP 2

STEP 3

Now, substitute log⁡7(53)\log{7}(5^3)log7(53) back into the original equation.log⁡7(x)+log⁡7(53)=5\log{7}( x)+\log{7}(5^3)=5log7(x)+log7(53)=5

STEP 4

We can use the property of logarithms that states the logarithm of the product of two numbers is equal to the sum of the logarithms of each number. This allows us to combine the two logarithms on the left side of the equation.log⁡7(4x⋅3)=\log{7}(4 x \cdot^3)=log7(4x⋅3)=

STEP 5

implify the inside of the logarithm.log⁡7(500x)=5\log{7}(500x)=5log7(500x)=5

STEP 6

To remove the logarithm, we can use the property of logarithms that states blog⁡b(x)=xb^{\log{b}(x)}=xblogb(x)=x. This gives us5=500x^5 =500x5=500x

STEP 7

To solve for xxx, we can divide both sides of the equation by500.x=75500x = \frac{7^5}{500}x=50075​

STEP 8

Finally, calculate the value of xxx.x=75500=16807500=33.614x = \frac{7^5}{500} = \frac{16807}{500} =33.614x=50075​=50016807​=33.614So the exact value of xxx is 33.61433.61433.614.

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Question Solve for the exact value of $x$ where $\log_7(4x) + 3\log_7(5) = 5$ .

Assumptions1. The base of the logarithm is7. . The logarithm of the product of two numbers is equal to the sum of the logarithms of each number.
3. The logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number.

Now, substitute $\log{7}(5^3)$ back into the original equation.
$\log{7}( x)+\log{7}(5^3)=5$

We can use the property of logarithms that states the logarithm of the product of two numbers is equal to the sum of the logarithms of each number. This allows us to combine the two logarithms on the left side of the equation.
$\log{7}(4 x \cdot^3)=$

implify the inside of the logarithm.
$\log{7}(500x)=5$

To remove the logarithm, we can use the property of logarithms that states $b^{\log{b}(x)}=x$ . This gives us $^5 =500x$

To solve for $x$ , we can divide both sides of the equation by500.
$x = \frac{7^5}{500}$

Finally, calculate the value of $x$ .
$x = \frac{7^5}{500} = \frac{16807}{500} =33.614$ So the exact value of $x$ is $33.614$ .