Solve equations of the form aˣ = b using logarithms — same base method and the general log/ln method — with step-by-step work showing every arithmetic detail.
Method 1 — Same Base: aˣ = aⁿ → x = n
Same Base Method: If you can rewrite both sides using the same base — for example, 8 = 2³ — then set the exponents equal and solve. This gives an exact integer answer with no calculator needed.
Log Method: When the same base trick isn't obvious, take log (or ln) of both sides. The Power Rule lets you bring the exponent down as a multiplier:
aˣ = b → log(aˣ) = log(b) → x · log(a) = log(b) → x = log(b) / log(a)
Both log base 10 and natural log (ln) give the same answer because the log base cancels in the ratio.
x = log_a(b) = log(b)/log(a) = ln(b)/ln(a)
Logarithms of the same base or natural log both work — choose whichever is cleaner. The ratio is what matters, not the base you pick.
For aˣ = b·cˣ: Divide both sides by cˣ first:
(a/c)ˣ = b → x = log(b) / log(a/c)
This works as long as a ≠ c (otherwise the bases cancel and there is no exponential to solve).
One-on-one Algebra 2 tutoring builds the intuition for when to use same-base versus logs — we work through your actual homework and tests so the strategy sticks.