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 change of base formula

The change of base formula for logarithms is $\log_b a=\frac{\log_c a}{\log_c b}$. We can use this to calculate $\log_5 15$ by changing the base to 10 (or natural logarithm, but base 10 is common for calculators). So $\log_5 15 = \frac{\log 15}{\log 5}$.

Step2: Calculate the logarithm values

Using a calculator, $\log 15\approx1.1761$ and $\log 5\approx0.6990$. Then $\frac{\log 15}{\log 5}=\frac{1.1761}{0.6990}\approx1.6826$.

Step3: Solve for x

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

Answer:

-1.317 (corresponding to the option: -1.317)