Logarithm Calculator
Finding the Hidden Exponent
A logarithm answers one direct question: to what power must the base be raised to reach the number? Enter a Base and a Number, press Calculate, and the exponent appears. Science and finance students meet logs in pH, decibels, and compound growth.
How Logs Are Solved
By definition, logb(x) = y means b^y = x. Any base converts cleanly through the identity logb(x) = ln(x)/ln(b), which the calculator applies under the hood.
A classic case: log2(8) = 3 because 2^3 = 8. In base 10, the log of 1000 is 3 since 10^3 = 1000, and the log of 316.23 rounds to 2.5.
Getting Reliable Results
Type the base and the target number, then press Calculate. Valid bases are positive and not equal to 1, and the Number must also stay positive.
Recall that logs grow slowly: each step of log base 10 multiplies the number by ten. Mixing up the Number and the base is the most frequent input mistake.
Compound growth gives the tool a practical workout. Say money grows 7 percent a year and you want the doubling time: set Base to 1.07 and Number to 2, press Calculate, and the result lands near 10.24, meaning about ten and a quarter years to double. Swap Number for 4 and the answer doubles to roughly 20.49, since quadrupling is two doublings. Note how the field roles matter: the base carries the growth factor while the Number is the target multiple, the reverse of how many people first type the inputs.
Frequently Asked Questions
Why does log2(8) equal 3?
Because 2 raised to the power 3 equals 8. The logarithm records that exponent, so log base 2 of 8 is exactly 3 with no approximation.
Which bases are allowed here?
Any positive number except 1 works, including 2, 10, and e. Base 1 fails because every power of 1 is still 1, so no exponent could exist.
How do I use base e?
Choose the number e, roughly 2.718, as your base for natural logarithms. The calculator then returns the natural log, standard in growth and decay problems.
Can the number be negative?
No. Real logarithms need a positive Number, since no real power of a positive base produces zero or a negative value. Complex results fall outside this tool.
What is the change of base formula?
It is logb(x) = ln(x)/ln(b), letting any logarithm be computed with natural logs. Divide the natural log of x by the natural log of b.