SOLUTION: use logarithms to solve each equation: 3(superscript x)= 5(superscript 5x-1)

Question 203386: use logarithms to solve each equation:
3(superscript x)= 5(superscript 5x-1)
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
3 (superscript x) is usually referred to as 3^x.
similarly 5 (superscript 5x-1) is usually referred to as 5^(5x-1).
^ is shift 6 on your keyboard.
-----
they should look like the following if I understood you correctly.
-----
as a general rule, if a = b, then log(a) = log(b), so if we let:
a = 3^x, and
b = 5^(5x-1), we get:
log(3^x) = log(5^(5x-1))
-----
by the rules of logarithms:
since we know, that log(3^x) is the same as x*log(3),
and we know that log(5^(5x-1)) is the same as (5x-1)*log(5), we get:
x*log(3) = (5x-1)*log(5)
-----
if we remove parentheses from the right side of the equation, we get:
x*log(3) = 5x*log(5) - log(5)
-----
if we add log(5) to both sides of this equation, and we subtract x*log(3) from both sides of this equation, we get:
log(5) = 5x*log(5) - x*log(3)
-----
if we separate the common factor x from the right side of the equation, we get:
log(5) = x*(5*log(5) - log(3))
which looks like this:
if we divide both sides of the equation by (5*log(5) - log(3)), we get:
x = log(5) / (5*log(5) - log(3))
which looks like this:
-----
now it's just a matter of finding log(5) and log(3) on the calculator and substituting in the equation to solve for x.
-----
since log(5) = .698970004
and log(3) = .477121255
we get:
x = .698970004 / (5*(.698970004) - .477121255)
which becomes:
x = .698970004 / 3.017728767
which becomes:
x = .231621215
-----
3^x = 3^(.231621215) = 1.289767428
and
5^(5x-1) = 5^(5*(.231621215) - 1)
which becomes:
5^(5x-1) = 5^(.158106076)
which becomes:
5^(5x-1) = 1.289767428
which means that:
3^x = 5^(5x-1)
because they both = 1.289767428
which means that the value of x is good, and your answer is:
x = .231621215

RELATED QUESTIONS
Solve for x to three significant digits. 3 Superscript 6x Baseline equals 5... (answered by Fombitz)

Solve for x to three significant digits. 3 Superscript 6x Baseline equals 5... (answered by Edwin McCravy)

5x.x.(8x-1)1/2=500 (answered by pannalal)

Rewrite the given equation so that it contains no logarithms. log ⁡ Subscript 6 (answered by Alan3354)

Need to solve the following - PLEASE HELP!!! 1. 4x^3/4 = 108 (exponent is 3/4)... (answered by Nate,stanbon)

Solve for x to three significant digits. 7 Superscript 6x Baseline equals 9... (answered by Alan3354)

The rectangle below has an area of y 4 + 1 1 y 2 + 3 0 y 4 +11y 2... (answered by ikleyn)

Suppose you are solving a trigonometric equation for solutions over the interval left... (answered by greenestamps)

I will try to write this correctly - 5^x-1 = 125^2x+3 ( the x-1 and 2x+3 should all... (answered by jim_thompson5910)