Write Exponential Equation in Logarithmic Form - Algebra-2

1. Fundamental Concepts

Core Relationship Between Exponential and Logarithmic Equations:
Exponential operations and logarithmic operations are inverse operations of each other. If an exponential equation is in the form $b^y = x$ (where $b>0$, $b\neq1$, and $x>0$), its corresponding logarithmic form is $\log_b x = y$.

In $b^y = x$, $b$ is the base, $y$ is the exponent, and $x$ is the power;

In $\log_b x = y$, $b$ remains the base (called the base of the logarithm), $x$ is the argument, and $y$ is the value of the logarithm.
Domain Restrictions:
For the logarithm $\log_b x$, the argument $x$ must be greater than 0, and the base $b$ must be greater than 0 and not equal to 1. This is derived from the properties of exponential functions (since any power of a positive number is positive, and if the base is 1, the power is always 1, which has no meaning for inverse operations).

Equivalence Conversion Principle:
The exponential form $b^y = x$ is completely equivalent to the logarithmic form $\log_b x = y$. Given one form, you can convert it to the other by following the rule: "keep the base unchanged, swap the positions of the exponent and the logarithm, and swap the positions of the power and the argument".
Common Special Bases:
When the base $b = 10$, the logarithm is called a common logarithm and can be abbreviated as $\lg x$, i.e., $\log_{10} x = \lg x$;
When converting, it should be noted that even for special bases, the equivalence conversion rule between exponents and logarithms must be followed, and the corresponding relationship between the base and the argument cannot be omitted.

Problem: Write the exponential equation $2^4 = 16$ in logarithmic form.

Validation: Substitute back into the original equation: $2^4 = 16$ ✓

Problem 1: Write the exponential equation $5^{x} = 125$ in logarithmic form.

Validation: Substitute back into the original equation: $5^3 = 125$ ✓

Problem 2: Convert the exponential equation to logarithmic form.

Steps: Note that the exponent is negative, but the rule still applies. Here, , , ;
Result: $\log_5 \frac{1}{5} = -1$.

Visual Strategy: Use a table to organize the base, exponent, and result for clarity.
Error-Proofing: Always check by substituting the logarithmic form back into the exponential form.
Concept Reinforcement: Practice converting between exponential and logarithmic forms with different bases.