Evaluating base-10 logarithms
A base-10 logarithm records the exponent needed to produce a positive number from 10. Powers of ten can be evaluated exactly, while other positive values use technology and the stated rounding
\[\log_{10}(10^n)=n,\quad 10^{\log_{10}(a)}=a\;(a>0)\]
- Interpret a logarithm as an exponent The base-10 logarithm is the exponent that produces the positive number \(\displaystyle 10^3=1000\Rightarrow\log_{10}(1000)=3\)
- Use the exact power when available Integer powers of ten give integer logarithms, including log₁₀(1) = 0. \(\displaystyle \log_{10}(1)=0,\quad \log_{10}(100000)=5\)
- Use technology for other positive values A positive value that is not an integer power of ten is evaluated with technology and rounded at the stated precision \(\displaystyle \log_{10}(200)\approx2.301\text{ to 3 d.p.}\)
- Reverse a logarithm with an antilogarithm If the logarithm is known, raise 10 to that power to recover the number \(\displaystyle \log_{10}(2.5)\approx0.398,\quad 10^{2.5}\approx316.228\)
Worked example
Evaluate the logarithm and the reverse logarithmic expression. Give the second answer to 3 decimal places
\[\log_{10}(10000),\quad10^{2.5}\]
- \(\displaystyle 10000=10^4\)Write the first number as a power of 10.
- \(\displaystyle \log_{10}(10000)=4\)The logarithm is the exponent in the power of 10.
- \(\displaystyle 10^{2.5}=316.227766\ldots\)Evaluate the antilogarithm with technology before rounding
- \(\displaystyle 10^{2.5}\approx316.228\)Rounding 316.227766... to 3 decimal places gives 316.228
Common mistakes
Counting digits instead of the exponent
✕\(\displaystyle \log_{10}(10000)=5\)✓\(\displaystyle 10000=10^4\Rightarrow\log_{10}(10000)=4\)
A logarithm is the exponent on 10, not the number of digits in the written valueReversing the operation
✕\(\displaystyle \log_{10}(N)=2\Rightarrow N=\log_{10}(2)\approx0.301\)✓\(\displaystyle \log_{10}(N)=2\Rightarrow N=10^2=100\)
To reverse a logarithm, raise 10 to the stated logarithmic value rather than taking another logarithmTreating a non-power as exact
✕\(\displaystyle \log_{10}(200)=2\)✓\(\displaystyle \log_{10}(200)\approx2.301\text{ to 3 d.p.}\)
Two is the logarithm of 100, not 200; a non-power of ten needs a technology value at the requested precisionDifficulty levels
introEvaluate exact logarithms and antilogarithms of powers of 10.
standardCombine exact powers of 10 with technology logarithms at stated decimal places
challengeEvaluate antilogarithms at stated precision and distinguish the direction of the operation
examChoose a logarithm or antilogarithm value from exact and direction-based distractors
Questions about this topic
What does log₁₀(a) mean?+
It is the exponent n such that 10ⁿ = a, provided a is positive
Why is log₁₀(1) equal to 0?+
The zeroth power of any non-zero number is 1, so ten to the zeroth power is one and the base-10 logarithm of one is zero
How is an antilogarithm found?+
If the base-10 logarithm of N is L, raise ten to the power L to find N. Use technology and the stated rounding when L is not an integer
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, General Mathematics, chapter 9: Variation
In this chapter
Direct variation: write the relationshipDirect variation: find the constant and missing valuesDirect variation: solve a practical problemInverse variation: write the relationshipInverse variation: find the constant and missing valuesInverse variation: solve a practical problem9B Inverse variation: Identify direct or inverse variation from tables and graphs9C Data transformations: Identify direct or inverse variation from tables and graphsSquare and reciprocal data transformationsEvaluate logarithms and antilogarithmsOrders of magnitudeCompare quantities on a log scaleLogarithmic data transformationsFit and use square, reciprocal and logarithmic models