Introduction to Logarithms: Videos & Practice Problems

To solve exponential equations, we can use logarithms as the inverse operation. For example, if \( b^x = m \), it can be rewritten as \( \log_b(m) = x \). The natural logarithm, denoted as \( \ln \), is specifically for base \( e \). Key properties include \( \log_b(b) = 1 \) and \( \log_b(1) = 0 \). Understanding these concepts allows for evaluating logs without a calculator, enhancing problem-solving skills in mathematics.

Logarithms Introduction

When solving polynomial equations, such as \(x^3 = 216\), the goal is to isolate \(x\) by performing the reverse operation. In this case, taking the cube root of both sides yields \(x = \sqrt[3]{216}\). However, when the variable \(x\) is in the exponent, as in \(2^x = 8\), we need to determine how many times \(2\) must be multiplied by itself to equal \(8\). This leads us to the conclusion that \(x = 3\).

For more complex equations like \(2^x = 216\), instead of multiplying \(2\) repeatedly, we can utilize logarithms. The logarithm is the inverse operation of exponentiation. To isolate \(x\), we take the logarithm of both sides, specifically using the same base as the exponential. Thus, we write:

\( \log_2(2^x) = \log_2(216) \

Since the logarithm and the exponential share the same base, this simplifies to:

\(x = \log_2(216)\)

This expression is in logarithmic form, which indicates the power to which the base \(2\) must be raised to yield \(216\). Understanding this relationship allows us to convert between exponential and logarithmic forms effectively.

To convert from exponential to logarithmic form, we start with the base of the exponent. For example, from \(3^x = 81\), we derive:

\(x = \log_3(81)\)

Conversely, to convert from logarithmic to exponential form, we again start with the base. For instance, from \(x = \log_4(64)\), we can express it as:

\(4^x = 64\)

Let’s practice this conversion with a few examples. Starting with \(x = \log_5(800)\), we convert it to exponential form:

\(5^x = 800\)

For \( \log_2(16) = 4\), we convert it to:

\(2^4 = 16\)

Lastly, for \(10^x = 4500\), the conversion yields:

\(x = \log_{10}(4500)\)

Notably, logarithms with base \(10\) are referred to as common logarithms and can simply be written as \(\log(4500)\). This common log has a dedicated button on calculators, making it convenient for evaluations.

Understanding these conversions and the properties of logarithms is essential for solving equations involving exponents and will greatly enhance your problem-solving skills in mathematics.

The Natural Log

In mathematics, logarithms are essential tools for solving exponential equations. Among the most commonly used logarithms are the common logarithm, which is base 10, and the natural logarithm, which is base e. The natural logarithm is denoted as ln, a notation derived from the words "natural log." This special notation helps distinguish it from other logarithmic bases.

When dealing with exponential equations, such as b^x = m, we can express them in logarithmic form: log_b(m) = x. This principle applies equally to the natural logarithm. For example, if we have e^x = m, we can rewrite this as log_e(m) = x, which simplifies to ln(m) = x. It is important to remember that log_e is almost always written as ln.

To illustrate this, consider the equation x = ln(17). This can be rewritten in exponential form as e^x = 17. Similarly, if we start with e^x = 4, we can convert this to logarithmic form as ln(4) = x. In both cases, we treat the base e just like any other base when converting between forms.

Understanding the natural logarithm and its properties allows for easier manipulation of equations involving exponential growth or decay, making it a vital concept in various fields, including mathematics, physics, and engineering.

Change the following logarithmic expression to its equivalent exponential form.
$\log_4x=5$

• A. $4^{x}=5$
• B. $x^4=5$
• C. $4^5=x$
• D. $5^4=x$

Answer: (C)

Change the following logarithmic expression to its equivalent exponential form.
$x=\log9$

• A. $x=9$
• B. $9^{x}=10$
• C. $1^{x}=9$
• D. $10^{x}=9$

Answer: (D)

Change the following exponential expression to its equivalent logarithmic form.
$3^{x}=7$

• A. $\log_37=x$
• B. $\log_73=x$
• C. $\log_3x=7$
• D. $\log7=3^{x}$

Answer: (A)

Change the following exponential expression to its equivalent logarithmic form.
$e^9=x+3$

• A. $\log\left(x+3\right)=9$
• B. $\ln\left(x+3\right)=9$
• C. $\ln9=x+3$
• D. $\log_9x=e^3$

Answer: (B)

Evaluate Logarithms

Understanding logarithms is essential as they serve as the inverse of exponential functions. This relationship allows us to evaluate logarithmic expressions without a calculator. For instance, consider the expression log2(∛2). By recognizing that the cube root of 2 can be expressed as an exponent, we rewrite it as log2(21/3). Utilizing the inverse property of logarithms, where logb(bx) = x, we find that log2(21/3) = 1/3.

Another important property is that logb(b) = 1. This means that the logarithm of a base to itself always equals one. For example, log2(2) = 1 and log10(10) = 1. Similarly, the logarithm of 1 in any base is always zero, as logb(1) = 0. This is because any number raised to the power of zero equals one, such as 20 = 1.

To further illustrate these properties, let’s evaluate a few examples. For log2(∛2), we rewrite it as log2(21/3), which simplifies to 1/3. For the natural logarithm of 1, ln(1), we recognize it as loge(1), which equals 0. In the case of log(10), this is equivalent to log10(10), yielding 1.

Lastly, consider log5(1/5). We can express 1/5 as 5-1, allowing us to rewrite the logarithm as log5(5-1). Applying the inverse property, we find that this simplifies to -1.

By mastering these properties and techniques, evaluating logarithmic expressions becomes straightforward, enabling you to solve problems efficiently without the need for a calculator.

Evaluate the given logarithm.
$\log_77^{0.3}$

• A. 1.79
• B. 7
• C. 1
• D. 0.3

Answer: (D)

Evaluate the given logarithm.
$\frac32\log1$

• A. $\frac32$
• B. 0
• C. 1
• D. 10

Answer: (B)

Evaluate the given logarithm.
$\log_9\frac{1}{81}$

• A. 81
• B. 2
• C. – 2
• D. 9

Answer: (C)

Here’s what students ask on this topic:

What is a logarithm and how is it related to exponents?

A logarithm is the inverse operation of an exponent. If you have an exponential equation like $b^x=m$, it can be rewritten in logarithmic form as ${\log }_b\left(m\right)=x$. This means that the logarithm gives you the power to which the base must be raised to get the number. For example, ${\log }_2\left(8\right)=3$ because $2^3=8$.

How do you convert between exponential and logarithmic forms?

To convert from exponential to logarithmic form, start with the base of the exponent. For example, if you have $b^x=m$, it converts to ${\log }_b\left(m\right)=x$. Conversely, to convert from logarithmic to exponential form, start with the base of the log. For example, ${\log }_b\left(m\right)=x$ converts to $b^x=m$.

What are the properties of logarithms?

Key properties of logarithms include:
1. ${\log }_b\left(b\right)=1$ because any number raised to the power of 1 is itself.
2. ${\log }_b\left(1\right)=0$ because any number raised to the power of 0 is 1.
3. ${\log }_b\left(b^x\right)=x$ because the log and the exponent cancel each other out.

What is the natural logarithm and how is it different from common logarithms?

The natural logarithm, denoted as $\ln$, is a logarithm with base $e$ (approximately 2.718). It is written as $\ln$ instead of ${\log }_e$. The common logarithm, denoted as $\log$, has a base of 10. Both follow the same properties but are used in different contexts. For example, $\ln \left(1\right)=0$ and $\log \left(1\right)=0$.

How do you evaluate logarithms without a calculator?

To evaluate logarithms without a calculator, use properties of logarithms. For example, ${\log }_2(2^{\frac{1}{3}}$ because the log and exponent cancel out. Similarly, $\ln \left(1\right)=0$ because any log of 1 is 0. For ${\log }_5\left(\frac{1}{5}\right)=-1$, rewrite $\frac{1}{5}$ as $5^{-}$ and use the property to get $-1$.