5^(x-1) = 3^x?

Question

please show your work so i can fully understand. solve for x

Actually, I deleted that question because right after I posted it, I figure it out. Dont accuse me, please. Thanks.

figured*--

Answer 1

I don't think I want to because you seem to like to delete your questions right after an answer is provided. This denies anyone the chance to learn from the answer and it also denies the answerer the chance to earn 'best answer' for their hard work!

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Edit: Okay, I'm going to give you one more chance, but please don't delete this question too.

METHOD 1:

5^(x - 1) = 3^x

Take the log[base 5] of both sides:
x - 1 = log_5(3^x)

On the right hand side, use the change of base rule
log_a (x)= log(x) / log(a). Use a base of 3.

x - 1 = log_3(3^x) / log_3(5)

x - 1 = x / log_3(5)

Use the change of base rule again:
x - 1 = x / [log (5) / log(3) ]

Invert the denominator and put it in the numerator.
x - 1 = x log(3) / log(5)

Multiply both sides by log(5)
log(5) x - log(5) = log(3) x

Subtract log(3) x from both sides:
log(5) x - log(3) x = log(5)

Factor out an x:
[log(5) - log(3)] x = log(5)

Solve for x:
x = log(5) / [ log(5) - log(3) ]

x ≈ 3.1506601

METHOD 2:

Take the log of both sides:

log(5^(x-1)) = log(3^x)

Using this rule of logs log(a^k) = k log(a), rewrite as:

(x - 1) log(5) = x log(3)

Solve for x:

x log(5) - log(5) = x log(3)
x [ log(5) - log(3) ] = log(5)
x = log(5) / [ log(5) - log(3) ]

x ≈ 3.1506601

Same result.

P.S. You lose additional points when you delete a question. If you have already figured it out, then just add a follow-up note and ask for someone to check it, or something...

Answer 2

(x-1) ln 5 = x ln 3
1.465x - 1.465 = x
0.465x = 1.465
x = 3.15