Math Snap

STEP 1

What is this asking?
We're adding two logs with the same base, so we need to combine them into a single log and simplify!
Watch out!
Don't forget the log properties, and make sure your final answer has a coefficient of 1 in front of the log.

STEP 2

1. Combine the logs.
2. Simplify the argument.

STEP 3

When we add logs with the same base, we can multiply their arguments.
It's like this: $\log_b(m) + \log_b(n) = \log_b(m \cdot n)$.
Here, our base is 9, and our arguments are $5x^6$ and $4x$.
So, let's multiply!

STEP 4

$$\log_9(5x^6) + \log_9(4x) = \log_9(5x^6 \cdot 4x)$$ We're doing this because the sum of logs with the same base is equal to the log of the product of the arguments.

STEP 5

Inside the log, we have $5x^6 \cdot 4x$.
Let's multiply the coefficients $5$ and $4$ first. $5 \cdot 4 = 20$, so we now have $20x^6 \cdot x$.

STEP 6

Now, let's multiply the variables.
Remember that $x$ is the same as $x^1$.
When we multiply variables with the same base, we add the exponents.
So, $x^6 \cdot x^1 = x^{6+1} = x^7$.

STEP 7

Putting it all together, we get $20x^7$.
So, our combined and simplified log is $\log_9(20x^7)$.

Our final answer is $\log_9(20x^7)$.
We have a single log with a coefficient of 1, just like the problem asked for!