How do you solve ${(\frac{3}{4})}^{x} = \frac{27}{64}$ $(3/4)^x=27/64$ ?

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Answer 1 · 2018-07-03T14:00:20

$x = 3$ $color(blue)(x=3)$

We could solve this by the use of logarithms in the following way:

Taking logs of both sides. It dosen't matter which base you use as long as you use the same on both sides.

$x ln (\frac{3}{4}) = ln (\frac{27}{64})$ $xln(3/4)=ln(27/64)$

$x = \frac{ln (\frac{27}{64})}{ln (\frac{3}{4})} = 3$ $x=(ln(27/64))/(ln(3/4))=3$

This method requires the use of a calculator or tables. We can solve this without these:

Notice we can write:

$27 = 3^{3}$ $27=3^3$

$64 = 4^{3}$ $64=4^3$

$:.$

$(3/4)^x=(3^3)/(4^3)$

By the laws of indices:

$(3^3)/(4^3)=(3/4)^3$

$:.$

$(3/4)^x=(3/4)^3$

Since both bases are the same, both exponents are equal:

$:.$

$x=3$

How do you solve (3/4)^x=27/64(34)x=2764(3/4)^x=27/64?