Solve for n: `log_5 5^(6n+1) = 13`

Expert Answers
stepherson eNotes educator| Certified Educator

log base answer equals exponent

What does that mean in this problem?

If we have `log_2x=3` then`2^3=x` 

`2^3 = x`


If we have`log_3x=4` then:

`3^4 = x`


In this problem we have `log_5(6n+1) = 13`

We can write that as:

`5^13 = 5^(6n+1)`

Since the base of the exponents are equal (5=5), the exponents must be equal.

13 = 6n+1

We now want our variable,`n` alone on one side and a number on the other side. Our variable is on one side, but it is not alone. Let’s subtract “1" from both sides.

13 - 1 = 6n + 1 - 1

12 = 6n

We want `n` not `6n` so let’s divide both sides by`6`

`12/6 = 6n/6`


aishukul | Student

To solve this, first change the log into exponential form. Setting it into exponential form makes it easier and simpler to solve:


Since the bases are the same number, they can cancel out and you are left with:



n=2. This is the answer. 

atyourservice | Student

`log_5 5^(6n+1) = 13`

We know that log = 13 so we set it up in exponential form (this will make the problem way easier):

`b^y = x`

`5^13 = 5^(6n + 1)`

the base is the same (5), so we don't need to heed it. We are solving for n so we set the exponents equal to each other:

`13 = 6n + 1`

subtract 1 from both sides:

`12 = 6n`

now divide by 6 to get n alone:


to check:

`log_5 5^((6)(2)+1)`

`log_5 5^(12+1)`

`log_5 5^13`

we can use `log (x) / log(b) ` to get the answer:

`log (5^13) / log (5) =13`