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Solve equations with variables in the exponent using logarithms or same-base methods
For b^x = r, take log: x·log(b) = log(r), so x = log(r)/log(b)
Use when bases cannot be made the same
If a^m = a^n, then m = n. Set exponents equal.
Use when both sides have or can be rewritten with the same base
For b^(2x) + cb^x + d = 0, let u = b^x to get u² + cu + d = 0
Solve quadratic, then solve each b^x = u for x
For 4^x = 8^(x-1), rewrite as (2²)^x = (2³)^(x-1), then solve 2x = 3(x-1)
Look for bases that are powers of each other
For b^x = r, take the logarithm of both sides: log(b^x) = log(r). Use the power rule to get x·log(b) = log(r), then divide: x = log(r) / log(b). You can use any logarithm base.
Use the same-base method when both sides have the same base, or can be rewritten with the same base. For example, 4^x = 2^(x+2) can be rewritten as (2²)^x = 2^(x+2), then 2^(2x) = 2^(x+2), so 2x = x + 2.
An equation like 4^x - 5·2^x + 4 = 0 where you can substitute u = 2^x to get u² - 5u + 4 = 0. Solve the quadratic, then solve each resulting exponential equation.
Yes! Unlike logarithmic equations (which require positive arguments), exponential equations can have negative, zero, or positive solutions. All real numbers are valid solutions as long as they satisfy the equation.
For equations with e (like e^x = 5), take the natural logarithm: ln(e^x) = ln(5). Since ln(e^x) = x, you get x = ln(5). The natural log is the inverse of the exponential function with base e.
Some exponential equations have no real solution. For example, 2^x = -5 has no real solution because exponentials with positive bases are always positive. However, it may have complex solutions.
Substitute your solution back into the original equation. For b^x = r with solution x, verify that b^x equals r (within rounding error). Use a calculator to compute the exponential with your solution.
Common errors: Forgetting to take log of BOTH sides, incorrectly applying the power rule, mixing up b^x = r (exponential) with log_b(x) = r (logarithmic), and arithmetic errors when working with logarithms.