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.
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they should look like the following if I understood you correctly.
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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))
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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)
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if we remove parentheses from the right side of the equation, we get:
x*log(3) = 5x*log(5) - log(5)
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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)
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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:
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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.
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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
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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
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