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Logarithmic Scales - Measured Value (Number) to Magnitude

Level 1

The math topic focuses on applying logarithmic scales to calculate magnitudes in different contexts such as pH levels, decibel levels, and Richter scale magnitudes from given measured values. Problems involve computing the pH of solutions based on hydrogen ion concentrations, determining the decibel (dB) level from sound energy measurements, and calculating earthquake magnitudes on the Richter scale from seismic wave measurements. Each problem gives a scenario with specific data and asks for the associated logarithmic magnitude.

Work on practice problems directly here, or download the printable pdf worksheet to practice offline.

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Logarithmic Scales - Measured Value (Number) to Magnitude Worksheet

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Logarithmic Scales - Measured Value (Number) to Magnitude
1
A LaTex expression showing \text{pH} = -\log{[\text{H} to the power of + ]}\\ \\ [\text{H} to the power of + ] = {0.01} \text{mL}/\text{mol}
What is the pH on the pH scale when the hydrogen ion concentration is 0.01 mL/mol?
a A LaTex expression showing \text{pH} = 3.5
b A LaTex expression showing \text{pH} = 2
2
A LaTex expression showing \text{dB} = 10\log{(\frac{\text{I}}{\text{I} sub 0 })}\\ \text{I} sub 0 = 10 to the power of -12 \text{W}/\text{m} to the power of 2 \\ \text{I} = {0.0001} \text{W}/\text{m} to the power of 2
What is the dB magnitude on the decibel scale when the sound energy is 0.0001 W/m^2?
a A LaTex expression showing \beta = 80 \text{dB}
b A LaTex expression showing \beta = 87 \text{dB}
3
A LaTex expression showing \text{M} = \log{(\frac{\text{I}}{\text{I} sub 0 })}\\ \text{I} sub 0 = 1\mu \text{m}\\ \text{I} = {1 multiplied by 10 to the power of 9 } \mu \text{m}
What is the magnitude on the Richter scale when the wave height is 1 x 10^9 micrometers?
a A LaTex expression showing \text{M} = 11
b A LaTex expression showing \text{M} = 9
4
A LaTex expression showing \text{pH} = -\log{[\text{H} to the power of + ]}\\ \\ [\text{H} to the power of + ] = {1 multiplied by 10 to the power of -6 } \text{mL}/\text{mol}
What is the pH on the pH scale when the hydrogen ion concentration is 1 x 10^-6 mL/mol?
a A LaTex expression showing \text{pH} = 6
b A LaTex expression showing \text{pH} = 8
5
A LaTex expression showing \text{pH} = -\log{[\text{H} to the power of + ]}\\ \\ [\text{H} to the power of + ] = {1 multiplied by 10 to the power of -8 } \text{mL}/\text{mol}
What is the pH on the pH scale when the hydrogen ion concentration is 1 x 10^-8 mL/mol?
a A LaTex expression showing \text{pH} = 6
b A LaTex expression showing \text{pH} = 8
6
A LaTex expression showing \text{dB} = 10\log{(\frac{\text{I}}{\text{I} sub 0 })}\\ \text{I} sub 0 = 10 to the power of -12 \text{W}/\text{m} to the power of 2 \\ \text{I} = {1 multiplied by 10 to the power of -6 } \text{W}/\text{m} to the power of 2
What is the dB magnitude on the decibel scale when the sound energy is 1 x 10^-6 W/m^2?
a A LaTex expression showing \beta = 67 \text{dB}
b A LaTex expression showing \beta = 60 \text{dB}
7
A LaTex expression showing \text{M} = \log{(\frac{\text{I}}{\text{I} sub 0 })}\\ \text{I} sub 0 = 1\mu \text{m}\\ \text{I} = {100} \mu \text{m}
What is the magnitude on the Richter scale when the wave height is 100 micrometers?
a A LaTex expression showing \text{M} = 2
b A LaTex expression showing \text{M} = 1
8
A LaTex expression showing \text{pH} = -\log{[\text{H} to the power of + ]}\\ \\ [\text{H} to the power of + ] = {1 multiplied by 10 to the power of -7 } \text{mL}/\text{mol}
What is the pH on the pH scale when the hydrogen ion concentration is 1 x 10^-7 mL/mol?
a A LaTex expression showing \text{pH} = 7.5
b A LaTex expression showing \text{pH} = 7