Question

Write the following using logarithms: (a) \(10^{2}=100\) (b) \(0.001=10^{-3}\) (c) \(\mathrm{e}^{-1.3}=0.2725\) (d) \(\mathrm{e}^{1.5}=4.4817\)

Short answer

Question: Rewrite the following equations using logarithms: (a) \(10^{2}=100\) (b) \(0.001=10^{-3}\) (c) \(\mathrm{e}^{-1.3}=0.2725\) (d) \(\mathrm{e}^{1.5}=4.4817\) Answer: (a) \(\log_{10}(100)=2\) (b) \(\log_{10}(0.001)=-3\) (c) \(\ln(0.2725)=-1.3\) (d) \(\ln(4.4817)=1.5\)

Step-by-step solution

(a) Rewrite the equation using logarithms

To write the equation \(10^{2}=100\) using logarithms, we simply take the logarithm of both sides, with base 10. This gives us: \(\log_{10}(10^2) = \log_{10}(100)\). Using the property of logarithms that states \(\log_{b}(b^x) = x\), we can simplify the left side of the equation: \(2=\log_{10}(100)\) So, the equation \(10^{2}=100\) can be written using logarithms as \(\log_{10}(100)=2\).

(b) Rewrite the equation using logarithms

To rewrite the equation \(0.001=10^{-3}\) using logarithms, we take the logarithm of both sides with base 10. This gives us: \(\log_{10}(0.001) = \log_{10}(10^{-3})\). Using the property of logarithms that states \(\log_{b}(b^x) = x\), we can simplify the right side of the equation: \(\log_{10}(0.001) = -3\) So, the equation \(0.001=10^{-3}\) can be written using logarithms as \(\log_{10}(0.001)=-3\).

(c) Rewrite the equation using logarithms

To rewrite the equation \(\mathrm{e}^{-1.3}=0.2725\) using logarithms, we take the natural logarithm (logarithm with base \(\mathrm{e}\)) of both sides. This gives us: \(\ln(\mathrm{e}^{-1.3}) = \ln(0.2725)\). Using the property of logarithms that states \(\ln(\mathrm{e}^x) = x\), we can simplify the left side of the equation: \(-1.3 = \ln(0.2725)\) So, the equation \(\mathrm{e}^{-1.3}=0.2725\) can be written using logarithms as \(\ln(0.2725)=-1.3\).

(d) Rewrite the equation using logarithms

To rewrite the equation \(\mathrm{e}^{1.5}=4.4817\) using logarithms, we take the natural logarithm (logarithm with base \(\mathrm{e}\)) of both sides. This gives us: \(\ln(\mathrm{e}^{1.5}) = \ln(4.4817)\). Using the property of logarithms that states \(\ln(\mathrm{e}^x) = x\), we can simplify the left side of the equation: \(1.5 = \ln(4.4817)\) So, the equation \(\mathrm{e}^{1.5}=4.4817\) can be written using logarithms as \(\ln(4.4817)=1.5\).

Key concepts

Properties of Logarithms

Logarithms are mathematical operations that determine how many times a number, called the base, is multiplied by itself to get another number. They possess several useful properties that simplify calculations and problem-solving in algebra.

• Product Rule: The log of a product is equal to the sum of the logs of the factors. Mathematically, \(\log_{b}(xy) = \log_{b}(x) + \log_{b}(y) \).


• Quotient Rule: The log of a quotient is the difference of the logs. It is written as \(\log_{b}\left(\frac{x}{y}\right) = \log_{b}(x) - \log_{b}(y) \).


• Power Rule: The log of a power is the exponent times the log of the base, given by \(\log_{b}(x^{n}) = n\log_{b}(x) \).

These rules help in transforming multiplication and division inside a logarithm into simpler addition and subtraction outside it. Understanding these properties of logarithms can greatly assist in solving and rewriting equations like in the example of \(10^2 = 100\) tackled earlier.

Natural Logarithm

The natural logarithm is a logarithm with the base \(e\), where \(e\) is an irrational number approximately equal to 2.71828. It is widely used in calculus and other branches of mathematics, especially when dealing with continuous growth models.

• Notation: It is often denoted as \(\ln(x)\), representing \(log_{e}(x)\).


• Simplifying Expressions: With natural logarithms, especially when you have expressions involving powers of \(e\), the property \(\ln(e^x) = x\) simplifies calculations drastically.


• Applications: Natural logarithms appear frequently in decay problems and are instrumental in the natural exponential function, \(e^x\).

When rewriting equations like \(\mathrm{e}^{-1.3} = 0.2725\) into logarithmic form, you would take the natural logarithm to get \(\ln(0.2725) = -1.3\), illustrating the power of using \(\ln\) for equations with base \(e\).

Logarithmic Equations

Logarithmic equations are equations that involve logarithms with unknown variables. Solving these equations often involves making use of properties of logarithms to simplify and isolate the variable.

• Converting Exponentials to Logarithms: One common approach is to rewrite the equation to log form, as seen in the example \(10^2 = 100\) being expressed as \(\log_{10}(100) = 2\).


• Using Logarithmic Properties: Applying properties like the power rule can help pull coefficients out of the log, which is useful in solving equations.


• Change of Base Formula: Occasionally, it helps to convert logs to a different base for easier computations, especially if a calculator is necessary. The change of base formula is \(\log_{b}(a) = \frac{\log_{c}(a)}{\log_{c}(b)}\).

In equations like \(\mathrm{e}^{1.5} = 4.4817\), converting to logarithmic form using natural logarithm simplifies to an easy-to-solve equation, demonstrating the value of knowing how to switch between exponential and logarithmic forms.