Exponential Equation Solver with Logarithm Steps
Solve a·b^(cx + d) + e = k for x step by step: isolate the power, take logarithms, solve the linear equation, and check the answer — with the no-solution case explained.
How it works
a·b^(cx + d) + e = k ⇒ b^(cx + d) = (k − e)/a ⇒ cx + d = log_b((k − e)/a) = ln((k − e)/a) / ln b ⇒ x = (log_b((k − e)/a) − d) / c
To solve an exponential equation, first isolate the power, then undo it with a logarithm of the same base (or with natural logs and the change-of-base rule log_b y = ln y / ln b). What remains is a linear equation in x. Because b^(anything) is always positive, there is no real solution when (k − e)/a is zero or negative.
Worked example
3 · 2^(x + 1) = 48: divide by 3 to get 2^(x + 1) = 16; take log base 2: x + 1 = log₂(16) = 4; so x = 4 − 1 = 3. Check: 3 · 2⁴ = 48. Solution x = 3.
Sources: OpenStax College Algebra 2e — 6.6 Exponential and Logarithmic Equations · Paul’s Online Math Notes — Solving Exponential Equations
Frequently asked questions
How do you solve an exponential equation with different bases?
Take logarithms of both sides. For 5^x = 2^(x + 3): x·ln 5 = (x + 3)·ln 2, so x = 3 ln 2 / (ln 5 − ln 2) ≈ 2.2694.
When can I solve without logarithms?
When both sides are powers of the same base: 2^(x + 1) = 16 = 2⁴ gives x + 1 = 4 directly.
Why is there no solution to 2^x = −8?
A positive base raised to any real power is positive, so it can never equal a negative number or zero.
Should I use ln or log?
Either — the answer is the same. ln y / ln b and log y / log b both equal log_b y.
Related calculators
This calculator is for educational purposes. Double-check important results before acting on them.