Question

question 4
score on last try: 2 of 3 pts. see details for more.
at least one scored part is incorrect. jump to first changable incorrect part.
next question get a similar question you can retry this question below
the power property of logarithms works as follows:
\\(\log_b(m^n) = n \cdot \log_b(m)\\)
use this property to rewrite each of the following with exponents removed. do not use the change of base formula. answers should be exact, not decimal approximations.
\\(\log_{25}(x^5) = \boxed{5\log_{25}(x)} \\)
\\(\ln(a^9) = \boxed{9\ln(a)} \\)
\\(\log_9(3^{m+4}) = \boxed{(m + 4)\log_9(3)} \\)
note: \\(\log_b(a)\\) can be written using an underscore like this: log_b(a).
question help: video
submit question

Explanation:

Step1: Recall Power Property of Logarithms

The power property of logarithms states that $\log_b(m^n) = n \cdot \log_b(m)$. This property applies to natural logarithms (where $b = e$, so $\ln(m^n) = n \cdot \ln(m)$) as well.

Step2: Apply the Property to $\ln(a^9)$

For the expression $\ln(a^9)$, we identify $b = e$, $m = a$, and $n = 9$. Using the power property of logarithms, we can rewrite $\ln(a^9)$ as $9 \cdot \ln(a)$. Wait, but the system marked this as incorrect? Wait, maybe there's a typo in the original problem? Wait, no, let's check again. Wait, the original problem says $\ln(a^9)$? Wait, maybe the variable is different? Wait, no, the user's problem shows $\ln(a^9)$ with the answer $9\ln(a)$ marked wrong. Wait, no, maybe I misread. Wait, no, the power property is correct. Wait, maybe the problem was $\ln(a^g)$? Wait, looking at the image, it's $\ln(a^g)$? Wait, the user's image shows $\ln(a^g)$ (maybe a typo for $a^9$? Wait, no, the box has $9\ln(a)$ with a cross. Wait, maybe the exponent is $g$? Wait, no, the user's text: "ln(a^g) = 9 ln(a) ×". Oh! Wait, the exponent is $g$, not $9$. So that's the mistake. So the correct application: for $\ln(a^g)$, using the power property, it should be $g \cdot \ln(a)$, not $9\ln(a)$. So the error was using $9$ instead of $g$. So the correct rewrite is $g\ln(a)$.

Answer:

For $\ln(a^g)$, using the power property of logarithms $\log_b(m^n)=n\log_b(m)$ (with $b = e$ for natural log), we rewrite $\ln(a^g)$ as $g\ln(a)$. So the correct answer is $g\ln(a)$.