Solve for n: `log_5 5^(6n+1) = 13`
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`
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.
`log_5 5^(6n+1) = 13`
`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:
we can use `log (x) / log(b) ` to get the answer:
`log (5^13) / log (5) =13`