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Solve the following logarithm problem for the positive solution for x. log_(x)25=(2)/(3) Answer:

Solve the following logarithm problem for the positive solution for x x .\newlinelog⁡x25=23 \log _{x} 25=\frac{2}{3} \newlineAnswer:

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Full solution

Q. Solve the following logarithm problem for the positive solution for x x .\newlinelog⁡x25=23 \log _{x} 25=\frac{2}{3} \newlineAnswer:
  1. Understand the logarithmic equation: Understand the logarithmic equation.\newlineThe equation log⁡x25=23\log_{x}25 = \frac{2}{3} means that xx raised to the power of 23\frac{2}{3} equals 2525. We can rewrite this equation in exponential form to find the value of xx.
  2. Convert to exponential form: Convert the logarithmic equation to exponential form.\newlineUsing the definition of a logarithm, we can write the equation as x23=25x^{\frac{2}{3}} = 25.
  3. Solve for x: Solve for x.\newlineTo find xx, we need to raise both sides of the equation to the reciprocal of 23\frac{2}{3}, which is 32\frac{3}{2}, to get rid of the fractional exponent on xx.\newline(x23)32=2532(x^{\frac{2}{3}})^{\frac{3}{2}} = 25^{\frac{3}{2}}
  4. Simplify the equation: Simplify the equation.\newlineWhen we raise a power to a power, we multiply the exponents. Therefore, (23)×(32)=1(\frac{2}{3}) \times (\frac{3}{2}) = 1, and we get:\newlinex1=2532x^1 = 25^{\frac{3}{2}}
  5. Calculate 253225^{\frac{3}{2}}: Calculate 253225^{\frac{3}{2}}. To calculate 253225^{\frac{3}{2}}, we first find the square root of 2525, which is 55, and then raise it to the power of 33. (2512)3=53=125(25^{\frac{1}{2}})^3 = 5^3 = 125
  6. Conclude the value of xx: Conclude the value of xx. Since x1=125x^1 = 125, we find that x=125x = 125.

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