Simplify $\frac{{{4}^{x+2}}-{{2}^{2x+1}}}{{{8}^{x}}\left( {{4}^{1-x}} \right)}$ and express it in

Question

Simplify $\frac{{{4}^{x+2}}-{{2}^{2x+1}}}{{{8}^{x}}\left( {{4}^{1-x}} \right)}$ and express it in the form $a\left( {{2}^{x-b}} \right)$ where $a$ and $b$ are integers.[3] Hence, solve for $x$ if $\frac{{{4}^{x+2}}-{{2}^{2x+1}}}{{{8}^{x}}\left( {{4}^{1-x}} \right)}=15$.[2]

Accepted Answer

(a) $7(2^{x-1})$, so $a=7$, $b=1$. (b) $x=1+\dfrac{\ln\dfrac{15}{7}}{\ln2}\approx2.10$.

Exponential and Logarithmic Functions - Equality of Indices

Exponential and Logarithmic Functions - Equality of Indices

Solution:

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