Question

question 6 of 10
which expressions are equivalent to the one below? check all that apply.
$log_5 125 + log_5 125$

a. $log 15625$

b. $log_5(5^6)$

c. $log_5 15625$

d. $6$

Explanation:

Step1: Simplify \(\log_{5}125\)

We know that \(5^3 = 125\), so by the definition of logarithms, \(\log_{5}125=\log_{5}(5^3) = 3\) (since \(\log_{a}(a^x)=x\) for \(a>0,a
eq1\)).

Step2: Calculate \(\log_{5}125+\log_{5}125\)

Substitute the value of \(\log_{5}125\) we found in Step 1. So \(\log_{5}125+\log_{5}125=3 + 3=6\).

Step3: Analyze Option A

Option A is \(\log15625\). This is a common logarithm (base 10), while our original expression has base 5. Let's check the value: \(\log(15625)\approx\log(1.5625\times 10^{4})=\log(1.5625)+4\approx0.1934 + 4 = 4.1934
eq6\), so A is not equivalent.

Step4: Analyze Option B

Option B is \(\log_{5}(5^{6})\). Using the property \(\log_{a}(a^x)=x\), we get \(\log_{5}(5^{6}) = 6\), which is equal to our result from Step 2.

Step5: Analyze Option C

Option C is \(\log_{5}15625\). Let's find what power of 5 gives 15625. We know that \(5^6=15625\) (since \(5^3 = 125\), \(5^6=(5^3)^2=125^2 = 15625\)). So \(\log_{5}15625=\log_{5}(5^6)=6\) (by \(\log_{a}(a^x)=x\)), which is equal to our result from Step 2.

Step6: Analyze Option D

We found in Step 2 that \(\log_{5}125+\log_{5}125 = 6\), so D is equivalent.

Answer:

B. \(\log_{5}(5^{6})\), C. \(\log_{5}15625\), D. \(6\)