Math  /  Algebra

QuestionEvaluate the expression. (Simplify your answer completely.) (a) log8(64)\log _{8}(64) \square (b) log7(49)\log _{7}(49) \square (c) log8(813)\log _{8}\left(8^{13}\right)

Studdy Solution

STEP 1

What is this asking? We need to figure out the values of a few logarithms.
Basically, we're trying to find out what power we need to raise the base to, to get the number inside the logarithm. Watch out! Don't mix up the base and the number we're taking the logarithm of.
Also, remember that logb(bx)=x\log_{b}(b^x) = x.
This is a super useful property!

STEP 2

1. Evaluate log8(64)\log_{8}(64)
2. Evaluate log7(49)\log_{7}(49)
3. Evaluate log8(813)\log_{8}(8^{13})

STEP 3

Alright, let's **rewrite** 64 as a power of 8!
We know that 88=648 \cdot 8 = 64, so we can write 64 as 828^2.

STEP 4

Now, we have log8(82)\log_{8}(8^2).
Since logarithms and exponents are inverse operations, the logarithm base 8 of 828^2 is simply **2**!
Think of it like this: what power do we raise 8 to, to get 828^2?
Well, 2 of course!

STEP 5

Let's **rewrite** 49 as a power of 7.
We know that 77=497 \cdot 7 = 49, which means 49=7249 = 7^2.

STEP 6

So, we have log7(72)\log_{7}(7^2).
Using the same logic as before, the answer is **2**!
We raise 7 to the power of 2 to get 727^2.

STEP 7

This one is already set up perfectly for us!
We have log8(813)\log_{8}(8^{13}).

STEP 8

What power do we need to raise 8 to in order to get 8138^{13}?
It's **13**!
So, log8(813)=13\log_{8}(8^{13}) = 13.
That's a big exponent, but the logarithm handles it like a champ!

STEP 9

(a) log8(64)=2\log_{8}(64) = 2 (b) log7(49)=2\log_{7}(49) = 2 (c) log8(813)=13\log_{8}(8^{13}) = 13

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