Logarithms as Inverse Exponents

A logarithm answers one question: "what exponent produces this number?" \(\log_b(x) = y\) means exactly the same thing as \(b^y = x\) — so a log is just the inverse of the exponential function with the same base.

By the end you'll be able to convert between log form and exponential form, evaluate logarithms by asking "what power?", and explain why the graphs of \(y=b^x\) and \(y=\log_b(x)\) are mirror images across the line \(y=x\).

Predict: with base b = 2, what is \(\log_2(8)\)? Think "2 to what power gives 8?" — then drag the value slider to 8 below and read the reflected point to check.

This plots \(y=b^x\) (blue) and its reflection across the dashed line y = x — that reflection is \(y=\log_b(x)\) (green), because a logarithm is exactly the inverse of the exponential: swap the roles of input and output and you get the other curve. Drag b to reshape both curves, and drag the value slider to slide a matched pair of points — one on each curve — showing that \(b^x=\text{value}\) and \(\log_b(\text{value})=x\) are the same fact seen from two directions.

\(2^{1.00} = 2.00\), so \(\log_2(2.00) = 1.00\) — the log answers "2 to what power gives 2.00?"

y = b^x reflected across \(y=x\) gives y = log_b(x)
y = b^x y = log_b(x) y = x (mirror line)

A logarithm just names an exponent: \(\log_b(x)\) is the power you raise base b to, to land on x (the argument) — which makes it the inverse of exponentiation.

Intuitive

Ask yourself "what power?" every time you see a log. \(\log_2(8)\) is asking "2 to what power gives 8?" — and since \(2^3=8\), the answer is 3. That's the whole trick: rewrite the log as the matching exponential question, then solve the exponential question instead, which is usually easier.

Formal

For base \(b>0\), \(b\neq 1\): \(\log_b(x) = y\) means exactly \(b^y=x\). Because exponentiating and logging with the same base undo each other, \(\log_b(b^x)=x\) and \(b^{\log_b(x)}=x\) for every valid x. Two bases get their own shorthand: \(\log_{10}(x)\) is written \(\log(x)\) (the common log), and \(\log_e(x)\) is written \(\ln(x)\) (the natural log). Because a positive base raised to any real power is always positive, the argument x must be positive — \(\log_b(0)\) and \(\log_b(\text{negative number})\) are both undefined.

Applied

Anywhere a quantity grows or decays by a constant factor — radioactive decay, earthquake magnitude, sound intensity in decibels, pH — logarithms let you ask "how many powers of b did this take?" instead of tracking huge or tiny numbers directly. If a population triples every year, \(\log_3(\text{growth factor})\) tells you directly how many years have passed.

Worked example

Evaluate \(\log_5(125)\) by converting to exponential form. Set the log equal to the unknown exponent: \(\log_5(125)=y\), so \(5^y=125\). Write 125 as a power of 5: \(125=5^3\). Match exponents: \(5^y=5^3\), so \(y=3\). Check: 5 to what power gives 125? The cube, since \(5\cdot5\cdot5=125\).

Your turn

Evaluate \(\log_2(1/8)\). Set it equal to the unknown exponent: \(\log_2(1/8)=y\), so \(2^y=1/8\). Write 1/8 as a power of 2 with a negative exponent: \(1/8 = 2^{-3}\). Match exponents: \(2^y=2^{-3}\), so \(y=\) ____

Reveal the answer

\(y = -3\), so \(\log_2(1/8)=-3\). Check: 2 to what power gives 1/8? A negative power, since raising 2 to a negative exponent flips it into a fraction: \(2^{-3}=1/2^3=1/8\). Set b = 2 and drag the value slider toward 0.1 on the interactive above to see the matching point sit at a negative x.

More info — why the log graph is the exponential graph, reflected

Every point \((x, b^x)\) on the exponential curve corresponds to a point \((b^x, x)\) on the log curve — the coordinates just swap. Swapping the coordinates of every point on a curve is exactly what reflecting across the line \(y=x\) does geometrically, which is why \(y=\log_b(x)\) always looks like \(y=b^x\) flipped over that diagonal. This is the same general pattern behind any function and its inverse, not just exponentials and logs. See the Math is Fun link in Dive deeper for more worked reflections.

Check your understanding

Question 1 of 4

Evaluate \(\log_4(64)\).

Question 2 of 4

For any valid base b (b > 0, b ≠ 1), what is \(\log_b(1)\)?

Question 3 of 4

Why is \(\log_b(-8)\) undefined for any valid base b?

Question 4 of 4

Since \(\log_b(x)\) is defined as the inverse of \(f(x) = b^x\), how must the graphs of \(y = b^x\) and \(y = \log_b(x)\) relate to each other?

Recap

  • \(\log_b(x)=y\) means exactly \(b^y=x\) — a logarithm names an exponent, "b to what power gives x?"
  • Logs and exponentials with the same base undo each other: \(\log_b(b^x)=x\) and \(b^{\log_b(x)}=x\).
  • The graph of \(y=\log_b(x)\) is the graph of \(y=b^x\) reflected across the line \(y=x\), since a function and its inverse always swap coordinates.
  • The argument x must be positive — \(\log_b(0)\) and logs of negative numbers are undefined — while \(\log_b(1)=0\) for every valid base.
  • \(\log_{10}(x)\) is written \(\log(x)\); \(\log_e(x)\) is written \(\ln(x)\).

Dive deeper

Sources

  • Logarithms as Inverse Exponents