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Logarithmic equations are equations that involve the logarithm of a variable or expression.
A logarithm is a way to express exponents. It tells us how many times we need to multiply a certain number, called the base, to get another number.
For example, in the logarithm \(\text{log}_b(x) = y\), it means that base \(b\) raised to the power of \(y\) equals \(x\).
In this equation, base 10 is used, which is the most common base for logarithms, often called the common logarithm.

• Logarithm base 10: \(\text{log}_{10}(x)\)
• Natural logarithm (base e): \(\text{log}_e(x)\) or \(\text{ln}(x)\)

To solve a logarithmic equation, we often need to convert it to its exponential form for easier manipulation.

Converting logarithmic equations to exponential form is a key step in solving them.
When we have a logarithmic equation like \(\text{log}_b(x) = y\), we can rewrite it in exponential form as \ b^y = x \. This conversion uses the definition of a logarithm.
In the example from the exercise, the logarithmic equation is given as \ -\text{log}_{10}(h) = p \. The first step is to remove the negative sign by multiplying both sides by -1, which gives us \ \text{log}_{10}(h) = -p \
Using the definition of logarithms, we can then convert this to exponential form:
\ 10^{-p} = h \
This shows how the logarithmic and exponential forms are interrelated, making it easier to work with and solve these equations.

The base of a logarithm is the number that is raised to a power.
In common logarithms, the base is 10. This is why these logarithms are often written as \(\text{log}(x)\) instead of \(\text{log}_{10}(x)\).
Base 10 logarithms are very useful in science and engineering because they can simplify complex calculations.

• For example, \(\text{log}_{10}(100) = 2\) because 10 raised to the power of 2 is 100.
• Similarly, \(\text{log}_{10}(1000) = 3\) because 10 raised to the power of 3 is 1000.

In the given exercise, the problem involves a base 10 logarithm, making it straightforward to convert to exponential form. Remembering how to translate between logarithmic and exponential forms is crucial for tackling these types of problems.