Math  /  Algebra

QuestionUse the properties of logarithms to expand the following expression as much as possible. Simplify any numerical expressions that can be evaluated without a calculator. log7(49x5)\log _{7}\left(49 x^{5}\right)

Studdy Solution

STEP 1

What is this asking? We need to rewrite this *single* logarithm as the *sum* of simpler logs and constants, using our logarithm tricks! Watch out! Don't forget the properties of logarithms, like how multiplication inside turns into addition outside!

STEP 2

1. Expand the logarithm using the product rule.
2. Simplify the logarithm with the base and power rule.

STEP 3

Alright, let's **expand** this logarithm!
Remember the product rule: logb(mn)=logb(m)+logb(n)\log_b(m \cdot n) = \log_b(m) + \log_b(n).
This means if we've got multiplication *inside* the logarithm, we can split it into *two* separate logs added together.
So, log7(49x5)\log_7(49x^5) becomes log7(49)+log7(x5)\log_7(49) + \log_7(x^5)!

STEP 4

Now, let's **simplify**!
Notice that 4949 is actually 727^2.
This is awesome because we can use the power rule: logb(bn)=n\log_b(b^n) = n.
This means if the *base* of the logarithm is the same as the *base* of the stuff *inside*, we can simplify it!
So, log7(49)=log7(72)=2\log_7(49) = \log_7(7^2) = 2.

STEP 5

We also have another power inside the log: x5x^5.
We can use the power rule in a slightly different way: logb(mn)=nlogb(m)\log_b(m^n) = n \cdot \log_b(m).
This means we can bring that exponent (55) down in *front* as multiplication.
So, log7(x5)\log_7(x^5) becomes 5log7(x)5 \cdot \log_7(x).

STEP 6

Putting it all together, we had log7(49)+log7(x5)\log_7(49) + \log_7(x^5).
We simplified log7(49)\log_7(49) to 22, and we simplified log7(x5)\log_7(x^5) to 5log7(x)5 \cdot \log_7(x).
So our **final simplified expression** is 2+5log7(x)2 + 5 \cdot \log_7(x)!

STEP 7

2+5log7(x)2 + 5 \cdot \log_7(x)

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