$4 \cdot 8^{b} = 53$ $\newline$ Find the value of `b`

Full solution

4 \cdot 8^{b} = 53

\newline

Find the value of `b`

Isolate term with exponent: Isolate the term with the exponent. $\newline$ To solve for $b$ , we need to isolate the term $8^b$ . We can do this by dividing both sides of the equation by $4$ . $\newline$ $4 \times 8^b = 53$ $\newline$ $(4 \times 8^b) / 4 = 53 / 4$ $\newline$ $8^b = 53 / 4$ $\newline$ $8^b = 13.25$
Take logarithm of both sides: Take the logarithm of both sides. $\newline$ To solve for the exponent $b$ , we can take the logarithm of both sides of the equation. We can use any logarithm, but it's most convenient to use the logarithm with base $8$ , which is log base $8$ , or the natural logarithm (ln) and then convert it. $\newline$ Let's use the natural logarithm: $\newline$ $\ln(8^b) = \ln(13.25)$
Apply power rule of logarithms: Apply the power rule of logarithms. $\newline$ The power rule of logarithms states that $\ln(a^x) = x \cdot \ln(a)$ . We can apply this rule to the left side of the equation. $\newline$ $b \cdot \ln(8) = \ln(13.25)$
Solve for b: Solve for b. $\newline$ To solve for b, we divide both sides of the equation by $\ln(8)$ . $\newline$ $b = \frac{\ln(13.25)}{\ln(8)}$ $\newline$ Now we can use a calculator to find the values of $\ln(13.25)$ and $\ln(8)$ . $\newline$ $b \approx \frac{1.2837}{2.0794}$ $\newline$ $b \approx 0.617$