Problem 39 - Contemporary Precalculus

Question

Prove that for any positive number \(c, \log c=\frac{\ln c}{\ln 10} .[\)Hint: We know that \(10^{\log c}=c\) (why?). Take natural logarithms on both sides and use a logarithm law to simplify and solve for log \(c .]\)

Step-by-Step Solution

Answer

Question: Prove that for any positive number \(c\), the equation \(\log c = \frac{\ln c}{\ln 10}\) holds true using the given hint: \(10^{\log c} = c\). Answer: \(\log c = \frac{\ln c}{\ln 10}\)

Step 1: Write down the given equation.

We are given the equation \(10^{\log c} = c\). Our goal is to prove that \(\log c = \frac{\ln c}{\ln 10}\).

Step 2: Apply natural logarithm to both sides

We will apply the natural logarithm (\(\ln\)) to both sides of the equation \(10^{\log c} = c\). This gives us: \(\ln (10^{\log c}) = \ln c\)

Step 3: Use the log power rule & log base change rule

The logarithm power rule states that \(\ln(a^b) = b\ln a\). According to this rule, we have: \(\log c \cdot \ln 10 = \ln c\) Now, we need to isolate \(\log c\): \(\log c = \frac{\ln c}{\ln 10}\)

Step 4: Conclusion

We have proved that for any positive number \(c\), the equation \(\log c = \frac{\ln c}{\ln 10}\) is true.

Key Concepts

Logarithms

A logarithm is an operation that tells us the exponent needed to reach a given number from a specified base. For example, when we write \( \log_b a = x \) it illustrates that the base \( b \) raised to the power \( x \) gives us the number \( a \). Logarithms have the basic property that \( \log_b (b^x) = x \) because the base \( b \) raised to the \( x \) perfectly reaches the number \( b^x \).

This idea extends into solving various exponential equations, where logarithms are a crucial tool. By definition, the common logarithm has the base 10, often written simply as \( \log a \) instead of \( \log_{10} a \) for convenience. Understanding logarithms is fundamental because these functions are not just mathematical constructs, they appear everywhere in science, engineering, and even in the measurement of sound and earthquake intensities.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm with the special base \( e \), where \( e \) is an irrational number approximately equal to 2.71828. This base is particularly convenient for calculus because of its unique properties in relation to the growth rates and areas under curves, which are areas of great interest in continuous mathematics and applied sciences.

Just as the common logarithm \( \log \) relates to base 10, the natural logarithm relates to growth and decay processes. It's the inverse operation of taking \( e \) to some power, meaning \( \ln(e^x) = x \). For instance, if you have the number \( y = e^x \), then taking the natural logarithm of \( y \) will give you \( x \) because \( \ln(y) = \ln(e^x) = x \).

Logarithm Power Rule

The logarithm power rule is a property that greatly simplifies the process of working with logarithms in expressions that involve exponents. This rule states that for any positive real number \( a \) and real number \( b \) and \( c \) the following is true: \( \log_b(a^c) = c \cdot \log_b(a) \).

What makes this rule so useful is that it allows you to move the exponent of the argument of a logarithm in front of the logarithm, turning a multiplicative process into an additive one. This property is handy when you're dealing with equations where the variable is in the exponent since it enables you to isolate and solve for the variable.

Log Base Change Rule

When dealing with logarithms of different bases, you might need to convert one logarithm into another base for simplification or comparison. This is where the log base change rule comes into play. It states that for any positive real numbers \( a \) and \( b \) (with \( b eq 1 \) and \( a > 0 \) ), and any positive base \( c \) (with \( c eq 1 \) ), the logarithm \( \log_c a \) can be converted to base \( b \) as follows: \( \log_c a = \frac{\log_b a}{\log_b c} \).

This rule effectively allows you to change the base of a logarithm by using logarithms of another base that may be more convenient for calculations. For example, it’s very common to use base 10 or base \( e \) logarithms because they readily available on calculators and in computer software. This feature of logarithms is crucial for solving the original exercise and many other mathematical problems involving different bases.