91Ó°ÊÓ

Logarithms are a way to express exponents. They tell us the power to which a number, known as the base, must be raised to produce a certain value. For example, in the logarithm \(\text{log}_{10}100\), the base is 10, and it tells us that 10 must be raised to the power of 2 to get 100. This can be written as \(10^2 = 100\).
In our exercise, we are dealing with common logarithms where the base is 10. Understanding logarithms helps in many fields like science and engineering because they transform multiplicative processes into additive ones, making calculations simpler.
Some important properties of logarithms to remember are:

• \(\text{log}_{10}(AB) = \text{log}_{10}(A) + \text{log}_{10}(B)\)
• \(\text{log}_{10}(A/B) = \text{log}_{10}(A) - \text{log}_{10}(B)\)
• \(\text{log}_{10}(A^C) = C \times \text{log}_{10}(A)\)

Isolating variables means manipulating an equation to get the variable you are solving for on one side by itself. This often requires a series of steps involving algebraic operations such as addition, subtraction, multiplication, and division.
In our problem, we start with the equation \(\text{log} E - 12.2 = 1.44M\). To isolate \(E\), we first need to move the constant -12.2 to the other side by adding 12.2 to both sides:
\(\text{log} E - 12.2 + 12.2 = 1.44M + 12.2\)
Which simplifies to:
\(\text{log} E = 1.44M + 12.2\).
Now that we have isolated the logarithmic term, we can focus on eliminating the logarithm to solve for \(E\). Always remember, isolating variables typically involves reversing the operations affecting the variable in question.

Exponential functions are the inverses of logarithmic functions. They describe situations where quantities grow exponentially. This growth could be things like population growth, radioactive decay, or interest in finance.
In our exercise, to solve for \(E\), we need to remove the logarithm by exponentiating. If \(\text{log} E = X\), then \(E = 10^X\). Using this property on our isolated equation:
\(\text{log} E = 1.44M + 12.2\)
We turn this into its exponential form:
\(E = 10^{1.44M + 12.2}\).
This gives the solution for \(E\). Exponential functions often result in rapid increases or decreases and are common in many real-world applications. Understanding how to manipulate and use these functions is crucial for problem-solving in mathematics and science.