Solve the following equation for c.
[tex]10^{7 c-3}=d[/tex]
c =
Answer (2)
To solve for c in the equation 1 0 7 c − 3 = d , take the logarithm of both sides. After applying logarithmic properties, isolate c to find that c = 7 l o g ( d ) + 3 . This final result provides the value of c in terms of d .
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Take the logarithm base 10 of both sides: lo g 10 ( 1 0 7 c − 3 ) = lo g 10 ( d ) .
Simplify using logarithm properties: 7 c − 3 = lo g 10 ( d ) .
Add 3 to both sides: 7 c = lo g 10 ( d ) + 3 .
Divide by 7 to solve for c : c = 7 lo g 10 ( d ) + 3 .
Explanation
Understanding the Problem We are given the equation 1 0 7 c − 3 = d and we want to solve for c . This involves using logarithms to isolate the variable c .
Applying Logarithms To isolate c , we first take the logarithm base 10 of both sides of the equation: lo g 10 ( 1 0 7 c − 3 ) = lo g 10 ( d ) Using the property of logarithms that lo g b ( b x ) = x , we simplify the left side: 7 c − 3 = lo g 10 ( d )
Isolating the Term with c Next, we add 3 to both sides of the equation to further isolate c :
7 c = lo g 10 ( d ) + 3
Solving for c Finally, we divide both sides by 7 to solve for c :
c = 7 lo g 10 ( d ) + 3 Thus, the solution for c is: c = 7 lo g 10 ( d ) + 3
Final Answer The solution to the equation 1 0 7 c − 3 = d for c is: c = 7 lo g 10 ( d ) + 3
Examples
Logarithmic equations are incredibly useful in various real-world scenarios, such as calculating the magnitude of earthquakes using the Richter scale, determining the pH levels of solutions in chemistry, and modeling population growth in biology. For instance, if we know the intensity ( d ) of an earthquake, we can use the formula c = 7 l o g 10 ( d ) + 3 (where the constants are adjusted to fit the Richter scale) to find a value related to its magnitude. This allows scientists to quantify and compare the strength of different earthquakes, helping in disaster preparedness and structural engineering.
