About Solve Exponential Equations
This calculator solves exponential equations of the form b^x = y by taking the logarithm of both sides, and then checks the answer by raising the base to it. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.
The page opens on 1.05^x = 2, which asks how many periods it takes for something growing 5 percent each period to double. The answer is x = ln 2 / ln 1.05, about 14.2066990829. At 7 percent it is about 10.2447683511, which is the idea behind the rule of 72.
The same method gives half-life times: for a decay factor 0.5 per step use base 0.5 and a target equal to the fraction left.
How to use Solve Exponential Equations
- Choose a mode: log of any base, ln, log10, log2, antilog or solve b^x = y.
- Enter the base (not needed for ln, log10 and log2) and the number. You can type fractions like 1/8, powers like 2^10 or the constant e.
- Read the exact or decimal answer at the top.
- Open the steps for the definition, the change of base and, when the number is an integer, the product and power rules applied to its prime factors.
- For an exponential equation such as 1.05^x = 2, choose Solve b^x = y.
Features
- Log of any base, with exact rational answers when the number is an exact power of the base.
- Natural log, common log and binary log as one-click modes.
- Antilogarithm: b raised to the power y, exact for integer exponents.
- Solve b^x = y, the form of compound interest, growth and half-life questions.
- Change-of-base steps using both ln and log10.
- Product and power rules shown on the prime factors of an integer argument.
- Related logs in bases e, 10 and 2, and a list of the logarithm rules.
- Plain explanations of domain errors.
Solving b^x = y
Take the log of both sides and divide.
b^x = y
x · log(b) = log(y)
x = log(y) / log(b)Worked examples
1.05^x = 2 gives x ≈ 14.2066990829
Doubling time at 5 percent per period.
5^x = 40 gives x ≈ 2.29202967422
ln 40 / ln 5.
Privacy: how it works
- Everything is calculated by code running in your browser tab. The numbers you enter are not sent to Ulkaa or to any other server, and there is nothing to sign in to.
Tips
- To check a log, raise the base to your answer. It should give back the number.
- log of a number between 0 and 1 is negative, and log of 1 is 0 in every base.
- For doubling time at a rate r per period, solve (1 + r)^x = 2 with the Solve mode.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Numbers are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Numbers are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
How do I solve an exponential equation?
Take the logarithm of both sides, bring the exponent down with the power rule, and divide: x = log(y) / log(b).
How long does money take to double?
Solve (1 + r)^x = 2. At 5 percent per year it is about 14.2 years. The rule of 72 estimates it as 72 divided by the percentage.
More Logarithm Calculator pages
- Logarithm CalculatorPick the logarithm you need, enter the base and the number, and get the answer with the change-of-base working.
- Natural Log CalculatorEnter a positive number to get ln(x), the logarithm to base e.
- Log Base 2 CalculatorEnter a positive number to get log2(x), how many times you can halve it.
- Log Base 10 CalculatorEnter a positive number to get log10(x), the common logarithm.
- Antilog CalculatorEnter a base and an exponent to get the antilogarithm, b raised to y.
- Change of Base CalculatorEnter a base and a number to see log_b(x) worked out with the change-of-base formula.