How to Solve Log Equations Without Calculator

Logarithmic Equation:

\[ x = b^y \quad \text{for} \quad \log_b(x) = y \]

Solution:

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1. What is a Logarithmic Equation?

A logarithmic equation is an equation that involves the logarithm of an expression containing a variable. The basic form is \(\log_b(x) = y\), which is equivalent to \(x = b^y\).

2. How to Solve Logarithmic Equations

To solve logarithmic equations without a calculator:

Convert the logarithmic equation to its exponential form
If the equation has multiple logarithms, use logarithm properties to combine them
Solve the resulting equation for the variable
Check your solution against the domain of the original equation

3. Properties of Logarithms

Key properties used in solving logarithmic equations:

\(\log_b(xy) = \log_b(x) + \log_b(y)\)
\(\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\)
\(\log_b(x^n) = n\log_b(x)\)
\(\log_b(b) = 1\)
\(\log_b(1) = 0\)

4. Using the Calculator

Instructions: Enter any two known values (x, b, or y) to calculate the third unknown value. The calculator uses the relationship \(x = b^y\).

Note: The base (b) must be positive and not equal to 1. The argument (x) must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What if my equation has natural log (ln)?
A: Natural log is just log with base e (≈2.71828). The same principles apply.

Q2: How do I solve equations with logarithms on both sides?
A: If \(\log_b(M) = \log_b(N)\), then M = N (as long as M, N > 0).

Q3: What about equations with different bases?
A: Use the change of base formula: \(\log_b(x) = \frac{\log_k(x)}{\log_k(b)}\) for any positive k ≠ 1.

Q4: Why must the argument of a log be positive?
A: Because you can't raise a positive number to any power and get a negative result or zero.

Q5: How do I check my solution?
A: Plug your solution back into the original equation to verify it works.