Problem 8 - Calculus and its applications

Question

Write an equivalent exponential equation. $$ -\log _{10} h=p $$

Step-by-Step Solution

Answer

The equivalent exponential equation is \(h = 10^{-p}\).

Step 1: Understand the Logarithmic Equation

The given equation is \[-\log_{10} h = p\] This implies that the negative of the logarithm base 10 of \(h\) is equal to \(p\). Our goal is to find an equivalent exponential expression.

Step 2: Eliminate the Negative Sign

We start by eliminating the negative sign from the equation to work with a straightforward logarithm:\[\log_{10} h = -p\]

Step 3: Convert Logarithmic to Exponential Form

A logarithmic equation \(\log_b a = c\) can be rewritten in exponential form as \(b^c = a\). Using this property, we convert \(\log_{10} h = -p\) to its exponential form:\[h = 10^{-p}\]

Step 4: Finalize the Equivalent Exponential Equation

The equivalent exponential expression for the given logarithmic equation is:\[h = 10^{-p}\] This restates the original logarithmic relationship in exponential terms.

Key Concepts

Logarithmic Equations

Logarithmic equations are mathematical expressions that involve the logarithm of a variable, often set equal to another expression. These equations allow us to find how many times a specific number, known as the base, must be multiplied by itself to achieve another number. The general form of a logarithmic equation is \( \log_b a = c \), where \( b \) is the base of the log, \( a \) is the argument, and \( c \) is the result of the logarithm.

In the original exercise, we have \( -\log_{10} h = p \). When dealing with logarithmic equations, the first step can often involve simplifying the expression by removing any negative signs, which makes subsequent steps clearer. This elimination process gives \( \log_{10} h = -p \).

Remember, solving logarithmic equations often involves transforming them into an exponential equation, which can sometimes be a simpler form for interpretation or further calculation, either for manual solving or computational use.

Base 10 Logarithms

Base 10 logarithms, also known as common logarithms, are logarithms where the base \( b \) is 10. These are widely used in many fields such as science and engineering, especially because computer and scientific calculators often include a dedicated button for \( \log_{10} \). Using base 10 logs, expressions become more relatable when measuring quantities that are powers of ten, like the Richter scale for earthquakes or pH in chemistry.

In the context of the original exercise, the logarithmic equation involved \( \log_{10} h \). The significance of a base 10 log here is that it simplifies the conversion process: any value \( a \) written as \( 10^c \) is straightforwardly the inverse operation detailed in the resultant exponential equation. Here, the transformation showed \( 10^{-p} \) as the exponential form of \( \log_{10} h = -p \). This outcome tells us that \( h \) is equal to ten raised to the power of negative \( p \), providing a clear, concise representation of the relationship.

Exponential Form

The exponential form of an equation can offer a clearer picture by showing the direct result of raising a base to the power of an exponent. This form is handy when you're dealing with equations originating from logarithms because it often simplifies solving or visualizing the relationships between involved quantities.

For logarithmic equations such as \( \log_b a = c \), converting this into exponential form involves rewriting it as \( b^c = a \). This method (exponential form) directly translates the log equation into the power relationship it represents.

• It shows the base (\( b \)) of the logarithm raised to the exponent (\( c \)), resulting in the argument (\( a \)).

This simple rearrangement from the original equation \( \log_{10} h = -p \) gives us \( h = 10^{-p} \).

In such transformations, exponential form not only simplifies the equation but also serves as a vital tool in contexts where growth patterns are evaluated, such as investment growth over time, population increase, or radioactive decay computations.