Question 1189074: B. The strength of an earthquake is calculated using Richter's formula M= log (A/10^-6) where A is the seismograph amplitude of the earthquake measured in meters.
1. If the earthquake has a seismograph amplitude ot 10^-3 m, find the magnitude of the earthquake.
2. The magnitude of the earthquake that hit the Central Visayas in 2013 is 7.2 on the Richter scale. What is the seismograph amplitude of the earthquake?
Computations and Answers:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the formula given is:
m = log (a / 10^-6) where a is the amplitude and m is the magnitude.
if the amplitude is 10^-3, then the formula becomes:
m = log(10^-3 / 10^-6) = log(10^-3+6)) = log(10^3) = 3
if m = 7.2, then 7.2 = log(a / 10^-6)
this becomes 7.2 = log(a) - log(10^-6) which becomes 7.2 = log(a) - -6 * log(10) which becomes 7.2 = log(a) + 6 * log(10) which becomes 7.2 = log(a) + 6.
subtract 6 from both sides of the eqution to get:
7.2 - 6 = log(a)
simplify to get:
1.2 = log(a)
this is true if and only if 10^1.2 = a
sove for a to get:
a = 15.84893192.
that should be the amplitude.
confirm by replacing a in the original equation by that to get:
7.2 = log(a / 10^-6) becomes 7.2 = log(15.84893192 / 10^-6) = 7.2.
this confirms the solution is correct.
some log properties that were used.
log(a/b_) log(a) - log(b)
y = log(x) if and only if 10^y = x
and:
10^x = y if and only if log(y) = x
log(x) means the same as log10(x).
log10(x) means the log of b to the base of 10.
log10 is the log function on your calculator.
the general equation is:
loga(b) = c if and only if a^c = b
in reverse, a^c = b if and only if loga(b) = c
there is also a log base conversion formula that says.
loga(x) = logb(x) / logb(a)
this is useful to convert logs of any base to the base of 10 which is the log function of your calculator.
for example:
log2(16) = y if and only if 2^y = 16
solve for y to get y = 4 because 2^4 = 16.
using the base conversion formula, you could also have done:
log2(16) = log(16)/log(2) = 4 by using the log function of your calculator.
let me know if you have any questions.
theo
here's some references.
https://www.chilimath.com/lessons/advanced-algebra/logarithm-rules/
https://www.storyofmathematics.com/logarithm-rules
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