How to Find an Antilog on a Simple Calculator

Title: How to Find an Antilog on a Simple Calculator Category: mathematics

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An antilog reverses a logarithm: if $\log_{10}(y)=x$, then $y=10^x$. In everyday use, “antilog” usually means the common antilog, with base 10. So the number you enter is the exponent, and the result is a power of 10.

This article shows the calculator keys to use, how to handle a negative or decimal exponent, and what you can do if your calculator has only basic arithmetic keys.

Use the 10^x key

Look for a key marked $10^x$. On many scientific calculators it is a secondary function of the LOG key, accessed by pressing SHIFT or 2nd first. Calculator layouts differ: some ask you to press the function, enter the exponent, and then press equals; others ask you to enter the exponent first and then select the function. Follow the display and your calculator’s key sequence.

For example, to find the antilog of 2.5, calculate $10^{2.5}$:

$$10^{2.5} \approx 316.227766$$

The result is about 316.23. You can check it by taking the common logarithm: $\log_{10}(316.227766)$ is approximately 2.5.

If your calculator has a general power key, often marked $x^y$ or ^, enter 10 as the base and 2.5 as the exponent. The exact key order varies by model.

Work with negative and decimal exponents

A decimal exponent does not mean you should move the decimal point in the exponent. Calculate the power as written. For example:

$$10^{-1.5}=\frac{1}{10^{1.5}}\approx 0.0316228$$

A negative exponent gives a reciprocal, so the answer is between 0 and 1. This is a useful check: a positive common-log exponent gives a value greater than 1, while a negative one gives a value between 0 and 1.

A fractional exponent can also be rewritten to help check a result. For the earlier example,

$$10^{2.5}=10^2\times 10^{0.5}=100\sqrt{10}\approx 316.227766.$$

This works because $10^{0.5}=\sqrt{10}$.

If the calculator has only basic keys

A simple four-function calculator can find integer powers by repeated multiplication. For example, the antilog of 3 is

$$10^3=10\times10\times10=1000.$$

For a negative integer exponent, take the reciprocal: $10^{-3}=1/1000=0.001$.

A calculator without a power or scientific-function key cannot directly evaluate every fractional power. If the exponent is a simple fraction, a square-root key may help: $10^{1.5}=10\sqrt{10}$, so find the square root of 10 and multiply by 10. Otherwise, use a printed antilog table, a calculator with a power function, or a reliable digital calculator. An antilog table may require separating the exponent into its characteristic and mantissa; follow the table’s instructions for placing the decimal point.

Antilog with a different base

For a valid logarithm base $b>0$ with $b\ne1$, the antilog in base $b$ of $x$ is $b^x$. For example, the antilog of 2 in base 3 is $3^2=9$. If the calculator has only a common-log key but supports division, you can use the change-of-base identity:

$$b^x=10^{x\log_{10}(b)}.$$

For instance, the base-3 antilog of 2 is $10^{2\log_{10}(3)}=9$. A calculator with an $x^y$ key makes this calculation more directly.

Common mistakes

• Pressing LOG instead of its inverse. LOG calculates a logarithm; SHIFT/2nd plus LOG commonly calculates $10^x$.
• Using the wrong base. The ordinary “antilog” is base 10, but a stated base such as 2 or $e$ changes the calculation.
• Treating a negative exponent as a negative answer. $10^{-x}$ is positive; it equals $1/10^x$.
• Assuming every calculator uses the same key order. Check the display or manual, especially when entering the exponent and selecting $10^x$.
• Rounding too early. Keep several digits during intermediate steps, then round the final result to the precision you need.