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Logarithmic Scales - Measured Value Ratio to Magnitude Difference

Level 2

This math topic focuses on understanding logarithmic scales in various contexts, such as pH, decibel levels (dB), and the Richter scale. Each problem requires students to calculate the magnitude difference between two measurements by using the logarithms of their ratios. It covers scenarios of measuring hydrogen ion concentrations, sound energy differences, and earthquake wave sizes, presenting real-world applications of logarithmic functions. The skills sharpened here include logarithmic computation, interpretation of logarithmic scales in practical situations, and solving problems featuring logarithmic relationships between measured values.

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

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Logarithmic Scales - Measured Value Ratio to Magnitude Difference Worksheet

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Logarithmic Scales - Measured Value Ratio to Magnitude Difference
1
A LaTex expression showing \text{pH} = -\log{[\text{H} to the power of + ]}\\ \\ \frac{[\text{H} to the power of + ] sub 2 }{[\text{H} to the power of + ] sub 1 } = 398
If a solution has 398 times the Hydrogen ion concentration as another what is their difference on the pH scale?
a A LaTex expression showing \text{pH} sub 2 - \text{pH} sub 1 = -2.6
b A LaTex expression showing \text{pH} sub 2 - \text{pH} sub 1 = -0.6
2
A LaTex expression showing \text{dB} = 10\log{(\frac{\text{I}}{\text{I} sub 0 })}\\ \\ \frac{\text{I} sub 2 }{\text{I} sub 1 } = 15.8
If a sound has 15.8 times the sound energy as another what is their difference on the decibel scale?
a A LaTex expression showing \beta sub 2 - \beta sub 1 = 12
b A LaTex expression showing \beta sub 2 - \beta sub 1 = 8
3
A LaTex expression showing \text{M} = \log{(\frac{\text{I}}{\text{I} sub 0 })}\\ \\ \frac{\text{I} sub 2 }{\text{I} sub 1 } = 79.4
If an earthquake has 79.4 times the wave size as another what is their difference on the Richter scale?
a A LaTex expression showing \text{M} sub 2 - \text{M} sub 1 = 0.9
b A LaTex expression showing \text{M} sub 2 - \text{M} sub 1 = 1.9
4
A LaTex expression showing \text{M} = \log{(\frac{\text{I}}{\text{I} sub 0 })}\\ \\ \frac{\text{I} sub 2 }{\text{I} sub 1 } = 1.26 multiplied by 10 to the power of 7
If an earthquake has 1.26 x 10^7 times the wave size as another what is their difference on the Richter scale?
a A LaTex expression showing \text{M} sub 2 - \text{M} sub 1 = 7.6
b A LaTex expression showing \text{M} sub 2 - \text{M} sub 1 = 7.1
5
A LaTex expression showing \text{dB} = 10\log{(\frac{\text{I}}{\text{I} sub 0 })}\\ \\ \frac{\text{I} sub 2 }{\text{I} sub 1 } = 631
If a sound has 631 times the sound energy as another what is their difference on the decibel scale?
a A LaTex expression showing \beta sub 2 - \beta sub 1 = 28
b A LaTex expression showing \beta sub 2 - \beta sub 1 = 29
6
A LaTex expression showing \text{M} = \log{(\frac{\text{I}}{\text{I} sub 0 })}\\ \\ \frac{\text{I} sub 2 }{\text{I} sub 1 } = 10,000
If an earthquake has 10,000 times the wave size as another what is their difference on the Richter scale?
a A LaTex expression showing \text{M} sub 2 - \text{M} sub 1 = 3.5
b A LaTex expression showing \text{M} sub 2 - \text{M} sub 1 = 4
7
A LaTex expression showing \text{pH} = -\log{[\text{H} to the power of + ]}\\ \\ \frac{[\text{H} to the power of + ] sub 2 }{[\text{H} to the power of + ] sub 1 } = 630,957
If a solution has 630,957 times the Hydrogen ion concentration as another what is their difference on the pH scale?
a A LaTex expression showing \text{pH} sub 2 - \text{pH} sub 1 = -5.3
b A LaTex expression showing \text{pH} sub 2 - \text{pH} sub 1 = -5.8
8
A LaTex expression showing \text{M} = \log{(\frac{\text{I}}{\text{I} sub 0 })}\\ \\ \frac{\text{I} sub 2 }{\text{I} sub 1 } = 251
If an earthquake has 251 times the wave size as another what is their difference on the Richter scale?
a A LaTex expression showing \text{M} sub 2 - \text{M} sub 1 = 1.4
b A LaTex expression showing \text{M} sub 2 - \text{M} sub 1 = 2.4