Full solution

Q. Solve for

x

\newline

6 = 7^x

\newline

x =

_____

Identify equation and question: Identify the equation and what is being asked. $\newline$ We are given the equation $6 = 7^x$ and need to find the value of $x$ .
Recognize exponential equation: Recognize that this is an exponential equation. $\newline$ In the equation $6 = 7^x$ , $7$ is the base raised to the power of $x$ .
Use logarithms to solve: Realize that the equation cannot be solved using elementary algebra since the variable is in the exponent. $\newline$ We need to use logarithms to solve for $x$ .
Apply logarithm to both sides: Apply the logarithm to both sides of the equation. $\newline$ We can use the natural logarithm (ln) for this purpose. $\newline$ $\ln(6) = \ln(7^x)$
Use power rule of logarithms: Use the power rule of logarithms to bring the exponent down. $\newline$ The power rule states that $\ln(a^b) = b \cdot \ln(a)$ . $\newline$ $\ln(6) = x \cdot \ln(7)$
Isolate the variable x: Isolate the variable $x$ . $\newline$ To solve for $x$ , divide both sides of the equation by $\ln($ $7$ ). $\newline$ $x = \frac{\ln($ $6$ )}{\ln( $7$ )}
Calculate value of x: Calculate the value of x using a calculator. $\newline$ $x \approx \frac{\ln(6)}{\ln(7)}$ $\newline$ $x \approx 0.8959$ (rounded to four decimal places)