Math / Algebra

QuestionSimplify. Enter the result as a single logarithm with a coefficient of 1. $\log _{9}\left(5 x^{6}\right)+\log _{9}(4 x)=$

Studdy Solution

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 $\log_9(20x^7)$ .

STEP 8

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

Was this helpful?

QuestionSimplify. Enter the result as a single logarithm with a coefficient of 1. $\log _{9}\left(5 x^{6}\right)+\log _{9}(4 x)=$

Studdy Solution

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 $\log_9(20x^7)$ .

STEP 8

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

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionSimplify. Enter the result as a single logarithm with a coefficient of 1. log⁡9(5x6)+log⁡9(4x)=\log _{9}\left(5 x^{6}\right)+\log _{9}(4 x)=log9​(5x6)+log9​(4x)=

Studdy Solution

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⋅n)\log_b(m) + \log_b(n) = \log_b(m \cdot n)logb​(m)+logb​(n)=logb​(m⋅n).Here, our base is **9**, and our arguments are **5x65x^65x6 and 4x4x4x**.So, let's **multiply**!

STEP 4

log⁡9(5x6)+log⁡9(4x)=log⁡9(5x6⋅4x)\log_9(5x^6) + \log_9(4x) = \log_9(5x^6 \cdot 4x)log9​(5x6)+log9​(4x)=log9​(5x6⋅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 5x6⋅4x5x^6 \cdot 4x5x6⋅4x.Let's multiply the **coefficients** 555 and 444 first. 5⋅4=205 \cdot 4 = 205⋅4=20, so we now have 20x6⋅x20x^6 \cdot x20x6⋅x.

STEP 6

Now, let's multiply the variables.Remember that xxx is the same as x1x^1x1.When we multiply variables with the same base, we **add** the exponents.So, x6⋅x1=x6+1=x7x^6 \cdot x^1 = x^{6+1} = x^7x6⋅x1=x6+1=x7.

STEP 7

Putting it all together, we get 20x720x^720x7.So, our combined and simplified log is log⁡9(20x7)\log_9(20x^7)log9​(20x7).

STEP 8

Our final answer is log⁡9(20x7)\log_9(20x^7)log9​(20x7).We have a single log with a coefficient of **1**, just like the problem asked for!

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionSimplify. Enter the result as a single logarithm with a coefficient of 1. $\log _{9}\left(5 x^{6}\right)+\log _{9}(4 x)=$

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.

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

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!

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

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 $20x^6 \cdot x$ .

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, $x^6 \cdot x^1 = x^{6+1} = x^7$ .

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!