Change of Base and Scale Explorer

Math Algebra • Exponential and Logarithmic Functions

Written by STEM Calculators Team Published December 17, 2025 Updated February 24, 2026

View all topics

Log base comparison

Side-by-side graphs of \( \log_b(x) \) • drag to pan • wheel/pinch to zoom

Left

x: 0, y: 0 sx: 60, sy: 35

Right

x: 0, y: 0 sx: 60, sy: 35

Click Calculate to see the change-of-base steps, a comparison table, and the graphs.

Logarithmic scales explorer

pH • decibels • “Richter-style” magnitude as base-10 logs

Scale:

pH:

\([\mathrm{H}^+]\) (mol/L):

Ready

Choose a scale and click Convert.

Rate this calculator

0.0 /5 (0 ratings)

Be the first to rate.

Your rating

Name (optional) Review (optional)

You can update your rating any time.

Theory: Change of Base and Logarithmic Scales

A logarithm answers: “what exponent gives this value?” If \(b>0\) and \(b\neq 1\), then \[ y=\log_b(x)\quad\Longleftrightarrow\quad b^y=x \qquad (x>0). \] All logarithm graphs have a vertical asymptote at \(x=0\), and every \(\log_b(x)\) passes through \((1,0)\).

1) Change-of-base formula

Any base can be converted to any other base using:

\[ \boxed{\log_b(x)=\frac{\ln(x)}{\ln(b)}=\frac{\log_a(x)}{\log_a(b)}}. \]

This is why calculators often use \(\ln\) internally: once you can compute \(\ln(x)\), you can compute \(\log_b(x)\).

2) How the base changes the “steepness”

The derivative explains how fast \(\log_b(x)\) rises:

\[ \frac{d}{dx}\log_b(x)=\frac{1}{x\ln(b)}. \]

At \(x=1\), the slope is \(1/\ln(b)\). So larger \(b\) gives a flatter curve near \((1,0)\).

3) Key points and intercepts

\(\log_b(1)=0\), so the x-intercept is always \(\boxed{(1,0)}\).
\(\log_b(b)=1\), so each base has a “marker” point \(\boxed{(b,1)}\).
A y-intercept does not exist because \(x=0\) is not in the domain.

4) Logarithmic scales (applications)

Many real-world scales are logarithmic: equal steps on the scale represent multiplying by a constant factor.

\[ \boxed{\mathrm{pH}=-\log_{10}\!\left([\mathrm{H}^+]\right)} \qquad\Longleftrightarrow\qquad [\mathrm{H}^+]=10^{-\mathrm{pH}}. \]

Decibels (intensity)

\[ \boxed{L=10\log_{10}\!\left(\frac{I}{I_0}\right)} \qquad\Longleftrightarrow\qquad \frac{I}{I_0}=10^{L/10}. \]

Magnitude (base-10 model)

\[ \boxed{M=\log_{10}\!\left(\frac{A}{A_0}\right)} \qquad\Longleftrightarrow\qquad \frac{A}{A_0}=10^{M}. \]

Frequently Asked Questions

What is the change-of-base formula for logarithms?

The change-of-base formula is log_b(x) = ln(x) / ln(b), and more generally log_b(x) = log_a(x) / log_a(b). It lets you compute logs in any base using a convenient base such as ln or log10.

How does changing the base affect the graph of log_b(x)?

All log_b(x) graphs pass through (1,0) and have a vertical asymptote at x=0, but the steepness changes with the base. Larger bases produce a flatter curve near x=1, while bases closer to 1 make the curve steeper.

How do I convert hydrogen ion concentration to pH and back?

For pH, the relationship is pH = -log10([H+]). Inverting it gives [H+] = 10^(-pH), so each increase of 1 pH unit corresponds to a tenfold decrease in [H+].

What does a decibel level mean in terms of intensity ratio?

For intensity, decibels use L = 10 log10(I/I0). Solving for the ratio gives I/I0 = 10^(L/10), so an increase of 10 dB corresponds to multiplying intensity by 10.

Rate this calculator

Frequently Asked Questions

Related calculators