Rewrite as an equation.

Rewrite <math><mstyle displaystyle="true"><msub><mi>log</mi><mrow><mn>25</mn></mrow></msub><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>5</mn></mrow></mfrac><mo>)</mo></mrow><mo>=</mo><mi>x</mi></mstyle></math> in exponential form using the definition of a logarithm. If <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> and <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> are positive real numbers and <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> does not equal <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> , then <math><mstyle displaystyle="true"><msub><mi>log</mi><mrow><mi>b</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>y</mi></mstyle></math> is equivalent to <math><mstyle displaystyle="true"><msup><mrow><mi>b</mi></mrow><mrow><mi>y</mi></mrow></msup><mo>=</mo><mi>x</mi></mstyle></math> .

Create expressions in the equation that all have equal bases.

Rewrite <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><msup><mrow><mn>5</mn></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mi>x</mi></mrow></msup></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mn>5</mn></mrow><mrow><mn>2</mn><mi>x</mi></mrow></msup></mstyle></math> .

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

Solve for <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> .

The variable <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> is equal to <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

The result can be shown in multiple forms.

Exact Form:

Decimal Form:

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