Questions: log3 x = log5(x^2)^5

Solution Steps

To solve the equation \(\log_{3} x = \log_{5}^{x^{2^{5}}}\), we need to understand the notation and properties of logarithms. The left side is a standard logarithm, while the right side seems to be using an unconventional notation. Assuming it means \(\log_{5} (x^{2^{5}})\), we can equate the two expressions and solve for \(x\). This involves using properties of logarithms, such as the power rule, and potentially converting the bases to solve the equation.

We start with the equation given in the problem:

\[ \log_{3} x = \log_{5} (x^{2^{5}}) \]

Assuming \(2^{5} = 32\), we can rewrite the right side as:

\[ \log_{3} x = \log_{5} (x^{32}) \]

Using the power rule of logarithms, we can simplify the right side:

\[ \log_{5} (x^{32}) = 32 \cdot \log_{5} x \]

Thus, the equation becomes:

\[ \log_{3} x = 32 \cdot \log_{5} x \]

We can apply the change of base formula to express both logarithms in terms of natural logarithms:

\[ \log_{3} x = \frac{\log x}{\log 3} \quad \text{and} \quad \log_{5} x = \frac{\log x}{\log 5} \]

Substituting these into the equation gives:

\[ \frac{\log x}{\log 3} = 32 \cdot \frac{\log x}{\log 5} \]

Assuming \(\log x \neq 0\), we can divide both sides by \(\log x\):

\[ \frac{1}{\log 3} = 32 \cdot \frac{1}{\log 5} \]

Cross-multiplying leads to:

\[ \log 5 = 32 \cdot \log 3 \]

Taking the exponent of both sides gives:

\[ x = 5^{32} \]

However, the solution from the Python output indicates that \(x = 1\) is also a solution.

Final Answer