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Logarithmic Scales - Ratio and Magnitude to Magnitude

Level 2

This math topic focuses on applying logarithmic scales across various scenarios, particularly with pH levels, decibels, and Richter scale calculations. The problems require understanding how to calculate changes in magnitude based on given ratios, requiring the use of logarithmic functions. Each question presents a situation where the concentration or intensity changes by a specified factor, and students must determine the new logarithmic scale value. The worksheet covers advanced logarithm functions, helping to deepen understanding of logarithmic relationships in practical contexts like chemistry, seismology, and acoustics.

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

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Logarithmic Scales - Ratio and Magnitude to Magnitude Worksheet

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Logarithmic Scales - Ratio and Magnitude to Magnitude
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 } = 1,000,000 \\ \text{pH} sub 1 = 9.9
If a solution has 1,000,000 times the Hydrogen ion concentration as one with a pH of 9.9 on the pH scale, what is it's pH?
a A LaTex expression showing \text{pH} sub 2 = 3.9
b A LaTex expression showing \text{pH} sub 2 = 4.9
2
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 } = 126 \\ \text{pH} sub 1 = 5.3
If a solution has 126 times the Hydrogen ion concentration as one with a pH of 5.3 on the pH scale, what is it's pH?
a A LaTex expression showing \text{pH} sub 2 = 3.2
b A LaTex expression showing \text{pH} sub 2 = 4.2
3
A LaTex expression showing \text{dB} = 10\log{(\frac{\text{I}}{\text{I} sub 0 })}\\ \\ \frac{\text{I} sub 2 }{\text{I} sub 1 } = 2,511,886 \\ \beta sub 1 = 30 \text{dB}
If a sound has 2,511,886 times the sound energy as one with a dB magnitude of 30 on the decibel scale, what is it's dB magnitude?
a A LaTex expression showing \beta sub 2 = 95 \text{dB}
b A LaTex expression showing \beta sub 2 = 94 \text{dB}
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 \\ \text{M} sub 1 = 6.2
If an earthquake has 1.26 times the wave size as one with a magnitude of 6.2 on the Richter scale, what is it's magnitude?
a A LaTex expression showing \text{M} sub 2 = 6.3
b A LaTex expression showing \text{M} sub 2 = 6.8
5
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 \\ \text{pH} sub 1 = 8.2
If a solution has 630,957 times the Hydrogen ion concentration as one with a pH of 8.2 on the pH scale, what is it's pH?
a A LaTex expression showing \text{pH} sub 2 = 3.9
b A LaTex expression showing \text{pH} sub 2 = 2.4
6
A LaTex expression showing \text{dB} = 10\log{(\frac{\text{I}}{\text{I} sub 0 })}\\ \\ \frac{\text{I} sub 2 }{\text{I} sub 1 } = 1.58 \\ \beta sub 1 = 127 \text{dB}
If a sound has 1.58 times the sound energy as one with a dB magnitude of 127 on the decibel scale, what is it's dB magnitude?
a A LaTex expression showing \beta sub 2 = 129 \text{dB}
b A LaTex expression showing \beta sub 2 = 135 \text{dB}
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 } = 199,526 \\ \text{pH} sub 1 = 12.2
If a solution has 199,526 times the Hydrogen ion concentration as one with a pH of 12.2 on the pH scale, what is it's pH?
a A LaTex expression showing \text{pH} sub 2 = 6.9
b A LaTex expression showing \text{pH} sub 2 = 8.9
8
A LaTex expression showing \text{M} = \log{(\frac{\text{I}}{\text{I} sub 0 })}\\ \\ \frac{\text{I} sub 2 }{\text{I} sub 1 } = 63.1 \\ \text{M} sub 1 = 2.4
If an earthquake has 63.1 times the wave size as one with a magnitude of 2.4 on the Richter scale, what is it's magnitude?
a A LaTex expression showing \text{M} sub 2 = 4.2
b A LaTex expression showing \text{M} sub 2 = 5.2