Math Snap

STEP 1

What is this asking?
We need to rewrite a single logarithm as the sum of simpler logarithms.
Watch out!
Remember the rules of logarithms, especially the product rule!

STEP 2

1. Expand the logarithm using the product rule.
2. Simplify the expression.

STEP 3

The product rule for logarithms states that the logarithm of a product is the sum of the logarithms of the factors.
Mathematically, this means $\log_b(m \cdot n) = \log_b(m) + \log_b(n)$.
This rule is super important because it lets us break down complicated logarithms into smaller, easier-to-manage pieces!

STEP 4

In our case, we have $\log_7(7x)$.
We can think of this as $\log_7(7 \cdot x)$.
Applying the product rule, we get:
$$\log_7(7 \cdot x) = \log_7(7) + \log_7(x)$$ See how we broke down the logarithm of the product $7x$ into the sum of the logarithms of $7$ and $x$?
Awesome!

STEP 5

Remember, $\log_b(a)$ asks the question: "To what power must we raise $b$ to get $a$?".

STEP 6

So, $\log_7(7)$ asks: "To what power must we raise $7$ to get $7$?".
Well, $7$ raised to the first power is $7$ ($7^1 = 7$), so $\log_7(7) = 1$!

STEP 7

Now, we can substitute this back into our expanded expression:
$$\log_7(7) + \log_7(x) = 1 + \log_7(x)$$

Our final answer is $1 + \log_7(x)$.
We've successfully expanded and simplified the logarithm!