What this calculator does
When the unknown sits in an exponent, no amount of ordinary algebra will free it. Logarithms exist precisely for this: taking the log of both sides brings the exponent down where it can be handled.
The arithmetic is then straightforward. Solving 5 times 2 to the x equals 160 divides out the 5 to leave 32, and since 2 to the fifth is 32, x comes to exactly 5.000.
The formula
The equation is divided by the leading coefficient, then the logarithm of both sides is taken. The exponent comes down as a multiplier and is isolated by dividing by the log of the base.
| Term | Meaning |
|---|---|
| Logarithm | The inverse of exponentiation, which brings an exponent down. |
| Base | The number being raised to the power, which must be positive and not 1. |
| Change of base | Any logarithm base works, since the ratio is what matters. |
The inputs explained
| Field | What to enter |
|---|---|
| a | The coefficient multiplying the exponential. Cannot be zero. |
| b (base) | The base. Must be positive and not equal to 1. |
| c | The target value. Dividing it by a must give a positive result. |
When to use it
Finding a doubling time
Asking when growth reaches a target is exactly this equation.
Solving decay problems
A base below 1 gives decay, and the same method applies.
Understanding what logarithms are for
This is the problem they were invented to solve.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does the base change the answer?
The same equation at three bases.
Questions
Why does the base have to be positive and not 1?
A negative base raised to a fractional power is not real, and a base of exactly 1 gives 1 for every exponent, so the equation either has no solution or infinitely many. Both cases break the method.
Does it matter which logarithm I use?
No. Natural log, base 10 or any other gives the same answer, because the result is a ratio of two logarithms and the base cancels. Calculators usually offer natural log and base 10, and either works.
What if c over a is negative?
There is no real solution. A positive base raised to any real power is always positive, so it can never equal a negative number however large the exponent.
How does this give a doubling time?
Set the target to twice the starting value and the base to one plus the growth rate. The resulting exponent is the number of periods to double, which is what the rule of 72 approximates.
For geometric growth over discrete steps, see the geometric sequence calculator. For compounding money, see the compound interest calculator.