Question

what is the approximate value of x in the equation below?
\log_{5}15 = x + 3
\bigcirc -2.523
\bigcirc -1.317
\bigcirc 2.880
\bigcirc 7.485

Explanation:

Step1: Recall logarithm change of base formula

The change of base formula for logarithms is $\log_b a=\frac{\log_c a}{\log_c b}$, where $c>0,c
eq1$. We can use this to calculate $\log_5 15$ with base 10 (common logarithm) or base $e$ (natural logarithm). Let's use base 10 here. So $\log_5 15=\frac{\log 15}{\log 5}$.

Step2: Calculate $\log 15$ and $\log 5$

We know that $\log 15\approx1.1761$ and $\log 5\approx0.6990$. Then $\log_5 15=\frac{1.1761}{0.6990}\approx1.6826$.

Step3: Solve for $x$

We have the equation $\log_5 15 = x + 3$. Substitute the value of $\log_5 15$ we found: $1.6826=x + 3$. Then $x=1.6826 - 3=-1.3174\approx - 1.317$.

Answer:

-1.317 (the option corresponding to -1.317)