Notes: `10^(3x-8)=2^(5-x)` Solve the equation.

To solve the equation: $\displaystyle{{10}}^{{{3}{x}-{8}}}={{2}}^{{{5}-{x}}}$ , we may take "ln" on both sides.

$\displaystyle{\ln{{\left({{10}}^{{{3}{x}-{8}}}\right)}}}={\ln{{\left({{2}}^{{{5}-{x}}}\right)}}}$

Apply natural logarithm property: $\displaystyle{\ln{{\left({{x}}^{{n}}\right)}}}={n}\cdot{\ln{{\left({x}\right)}}}$ .

$\displaystyle{\left({3}{x}-{8}\right)}{\ln{{\left({10}\right)}}}={\left({5}-{x}\right)}{\ln{{\left({2}\right)}}}$

Let $\displaystyle{10}={2}\cdot{5}$ .

$\displaystyle{\left({3}{x}-{8}\right)}{\ln{{\left({2}\cdot{5}\right)}}}={\left({5}-{x}\right)}{\ln{{\left({2}\right)}}}$

Apply natural logarithm property: $\displaystyle{\ln{{\left({x}\cdot{y}\right)}}}={\ln{{\left({x}\right)}}}+{\ln{{\left({y}\right)}}}$ .

$\displaystyle{\left({3}{x}-{8}\right)}{\left({\ln{{\left({2}\right)}}}+{\ln{{\left({5}\right)}}}\right)}={\left({5}-{x}\right)}{\ln{{\left({2}\right)}}}$

Distribute to expand each side.

$\displaystyle{3}{x}{\ln{{\left({2}\right)}}}+{3}{x}{\ln{{\left({5}\right)}}}-{8}{\ln{{\left({2}\right)}}}-{8}{\ln{{\left({5}\right)}}}={5}{\ln{{\left({2}\right)}}}-{x}{\ln{{\left({2}\right)}}}$

Isolate all terms with x's on one side.

$\displaystyle{3}{x}{\ln{{\left({2}\right)}}}+{3}{x}{\ln{{\left({5}\right)}}}-{8}{\ln{{\left({2}\right)}}}-{8}{\ln{{\left({5}\right)}}}={5}{\ln{{\left({2}\right)}}}-{x}{\ln{{\left({2}\right)}}}$

$\displaystyle+{8}{\ln{{\left({2}\right)}}}+{8}{\ln{{\left({5}\right)}}}\ldots$

To solve the equation: $\displaystyle{{10}}^{{{3}{x}-{8}}}={{2}}^{{{5}-{x}}}$ , we may take "ln" on both sides.

$\displaystyle{\ln{{\left({{10}}^{{{3}{x}-{8}}}\right)}}}={\ln{{\left({{2}}^{{{5}-{x}}}\right)}}}$

Apply natural logarithm property: $\displaystyle{\ln{{\left({{x}}^{{n}}\right)}}}={n}\cdot{\ln{{\left({x}\right)}}}$ .

$\displaystyle{\left({3}{x}-{8}\right)}{\ln{{\left({10}\right)}}}={\left({5}-{x}\right)}{\ln{{\left({2}\right)}}}$

Let $\displaystyle{10}={2}\cdot{5}$ .

$\displaystyle{\left({3}{x}-{8}\right)}{\ln{{\left({2}\cdot{5}\right)}}}={\left({5}-{x}\right)}{\ln{{\left({2}\right)}}}$

Apply natural logarithm property: $\displaystyle{\ln{{\left({x}\cdot{y}\right)}}}={\ln{{\left({x}\right)}}}+{\ln{{\left({y}\right)}}}$ .

$\displaystyle{\left({3}{x}-{8}\right)}{\left({\ln{{\left({2}\right)}}}+{\ln{{\left({5}\right)}}}\right)}={\left({5}-{x}\right)}{\ln{{\left({2}\right)}}}$

Distribute to expand each side.

$\displaystyle{3}{x}{\ln{{\left({2}\right)}}}+{3}{x}{\ln{{\left({5}\right)}}}-{8}{\ln{{\left({2}\right)}}}-{8}{\ln{{\left({5}\right)}}}={5}{\ln{{\left({2}\right)}}}-{x}{\ln{{\left({2}\right)}}}$

Isolate all terms with x's on one side.

$\displaystyle{3}{x}{\ln{{\left({2}\right)}}}+{3}{x}{\ln{{\left({5}\right)}}}-{8}{\ln{{\left({2}\right)}}}-{8}{\ln{{\left({5}\right)}}}={5}{\ln{{\left({2}\right)}}}-{x}{\ln{{\left({2}\right)}}}$

$\displaystyle+{8}{\ln{{\left({2}\right)}}}+{8}{\ln{{\left({5}\right)}}}$ $\displaystyle+{8}{\ln{{\left({2}\right)}}}$ $\displaystyle+{8}{\ln{{\left({5}\right)}}}$

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$\displaystyle{3}{x}{\ln{{\left({2}\right)}}}+{3}{x}{\ln{{\left({5}\right)}}}+{0}+{0}={13}{\ln{{\left({2}\right)}}}-{x}{\ln{{\left({2}\right)}}}+{8}{\ln{{\left({5}\right)}}}$

$\displaystyle{3}{x}{\ln{{\left({2}\right)}}}+{3}{x}{\ln{{\left({5}\right)}}}={13}{\ln{{\left({2}\right)}}}-{x}{\ln{{\left({2}\right)}}}+{8}{\ln{{\left({5}\right)}}}$

$\displaystyle+{x}{\ln{{\left({2}\right)}}}$ $\displaystyle+{x}{\ln{{\left({2}\right)}}}$

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$\displaystyle{4}{x}{\ln{{\left({2}\right)}}}+{3}{x}{\ln{{\left({5}\right)}}}={13}{\ln{{\left({2}\right)}}}-{0}+{8}{\ln{{\left({5}\right)}}}$

$\displaystyle{4}{x}{\ln{{\left({2}\right)}}}+{3}{x}{\ln{{\left({5}\right)}}}={13}{\ln{{\left({2}\right)}}}+{8}{\ln{{\left({5}\right)}}}$

Factor out common factor $\displaystyle{x}$ on the left side.

$\displaystyle{x}{\left({4}{\ln{{\left({2}\right)}}}+{3}{\ln{{\left({5}\right)}}}\right)}={13}{\ln{{\left({2}\right)}}}+{8}{\ln{{\left({5}\right)}}}$

Divide both sides by $\displaystyle{\left({4}{\ln{{\left({2}\right)}}}+{3}{\ln{{\left({5}\right)}}}\right)}$ .

$\displaystyle\frac{{{x}{\left({4}{\ln{{\left({2}\right)}}}+{3}{\ln{{\left({5}\right)}}}\right)}}}{{{4}{\ln{{\left({2}\right)}}}+{3}{\ln{{\left({5}\right)}}}}}=\frac{{{13}{\ln{{\left({2}\right)}}}+{8}{\ln{{\left({5}\right)}}}}}{{{4}{\ln{{\left({2}\right)}}}+{3}{\ln{{\left({5}\right)}}}}}$

$\displaystyle{x}=\frac{{{13}{\ln{{\left({2}\right)}}}+{8}{\ln{{\left({5}\right)}}}}}{{{4}{\ln{{\left({2}\right)}}}+{3}{\ln{{\left({5}\right)}}}}}$

Apply natural logarithm property: $\displaystyle{n}\cdot{\ln{{\left({x}\right)}}}={\ln{{\left({{x}}^{{n}}\right)}}}$

$\displaystyle{x}=\frac{{{\ln{{\left({{2}}^{{{13}}}\right)}}}+{\ln{{\left({{5}}^{{8}}\right)}}}}}{{{\ln{{\left({{2}}^{{4}}\right)}}}+{\ln{{\left({{5}}^{{3}}\right)}}}}}$