Textbook
For $b>0$, $b\ne1$, and $x>0$, the logarithm $\log_bx$ is the unique real number $y$ satisfying $b^y=x$. For example, $\log_28=3$ because $2^3=8$. The base specifies which exponent is sought.
The natural logarithm $\ln x=\log_ex$ uses $e\approx2.71828$. The meanings of $\log x$ and $\lg x$ depend on convention, so the base must be stated. For positive $x,y$, the identities are $\log_b(xy)=\log_bx+\log_by$ and $\log_bx=\ln x/\ln b$.
The product rule does not apply to addition. For example, $\log_2(4+4)=3$, whereas $\log_24+\log_24=4$. One accidental match does not prove an identity.
The product rule follows from adding exponents: if $x=b^u$ and $y=b^v$, then $xy=b^{u+v}$, so $\log_b(xy)=u+v$. The logarithm increases for $b>1$ and decreases for $0<b<1$. Output zero corresponds to input 1 because $b^0=1$. Negative logarithms are permitted: for $b>1$, inputs between zero and one produce them.
Definitions and further reading: Basic Analysis I.
In plain language
A logarithm recovers an exponent: moving from 1 through 2 and 4 to 8 requires three doublings. The base determines each step, so changing it changes the answer. Exponents may be fractional and, for inputs below one with a base greater than one, negative. A staircase suggests the direction, but the number 3 between consecutive powers of two is not an illegal stop.
Example
The illustrative calculation repeatedly doubles an initial 1 until reaching 8. The count of doublings becomes $\log_28$.
Logarithms: Finding the Exponent
One doubling
Doubling 1 gives 2: the exponent is 1.
Step 1 of 3: One doubling
Read all steps
One doubling
Doubling 1 gives 2: the exponent is 1.
- State
- 1current2current
Two doublings
Another doubling gives 4.
- State
- 1current2current4current
Three doublings
Reaching 8 takes three steps, so the logarithm is 3.
- State
- 1current2current4current8current
The counter has an initial state: before the first doubling, the value is 1 and the step count is zero. That explains $\log_2 1=0$, rather than 1. The reversed calculation verifies the answer: exponent 3 must produce $2^3=8$. Using base 8 instead gives exponent 1 for the same input. The input remains fixed, but the question about the base has changed.
Related concepts: Exponentiation.