Question

Express in logarithmic form. a) \(12^{2}=144\) b) \(8^{\frac{1}{3}}=2\) 101 c) \(10^{-5}=0.000\) d) \(7^{2 x}=y+3\)

Short answer

a) \[\log_{12} 144 = 2\]; b) \[\log_{8} 2 = \frac{1}{3}\]; c) \[\log_{10} 0.00001 = -5\]; d) \[\log_{7} (y+3) = 2x\]

Step-by-step solution

- Understand the Conversion

To convert from exponential form to logarithmic form, use the relation: If \[a^b = c\]it can be written as \[\log_{a} c = b\]

- Convert Part (a)

Given: \[12^2 = 144\] Convert to logarithmic form: \[\log_{12} 144 = 2\]

- Convert Part (b)

Given: \[8^{\frac{1}{3}} = 2\] Convert to logarithmic form: \[\log_{8} 2 = \frac{1}{3}\]

- Convert Part (c)

Given: \[10^{-5} = 0.00001\] Convert to logarithmic form: \[\log_{10} 0.00001 = -5\]

- Convert Part (d)

Given: \[7^{2x} = y + 3\] Convert to logarithmic form: \[\log_{7} (y + 3) = 2x\]

Key concepts

logarithms

Logarithms are a fundamental concept in mathematics, especially in precalculus and calculus. They help us solve problems involving exponential growth or decay by turning multiplicative relationships into additive ones. The logarithm of a number is the exponent to which the base must be raised to get that number. For example, if \(a^b = c\), then the logarithmic form is \(\log_{a} c = b\). Practicing with logarithms strengthens your understanding of exponentiation and prepares you for more advanced math topics.
Let's look at an example: Suppose we have \(2^3 = 8\). In logarithmic form, this becomes \(\log_{2} 8 = 3\). The base of the exponent (\(2\)), becomes the base of the logarithm, the result (\(8\)) becomes the argument of the logarithm, and the exponent (\(3\)) is the result of the logarithm.

exponential functions

Exponential functions are equations where the variable is in the exponent. They're essential for modeling a variety of real-world phenomena, such as population growth, radioactive decay, and financial interest. The general form of an exponential function is \(f(x) = a \cdot b^x\), where \(a\) is a constant and \(b\) is the base of the exponential function (typically greater than 0).
For example, the equation \(2^x = 8\) is an exponential function. To solve it, we can use logarithms to find \(x\). Converting it to logarithmic form, we get \(\log_{2} 8 = x\), and we already know that \(\log_{2} 8 = 3\). Thus, \(x = 3\).
When learning exponential functions, it’s helpful to graph them to see how they grow or decay. Growth functions (*\(b > 1\)*) increase rapidly, while decay functions (*\(0 < b < 1\)*) decrease but never quite reach zero.

logarithmic equations

Logarithmic equations involve variables inside logarithms and often require conversion to exponential form for solving. To achieve this, we utilize the property that the logarithm of a number is the exponent to which the base must be raised to yield that number: \(\log_{a} c = b\) means \(a^b = c\). This property allows us to switch between exponential and logarithmic forms easily.
Let’s look at an example from our exercise: \(7^{2x} = y + 3\). To convert this into logarithmic form, we identify the base (\(7\)), the exponent (\(2x\)), and the result (\(y + 3\)). Thus, it becomes \(\log_{7} (y + 3) = 2x\). This way, we can solve for \(x\) once we have the value of \(y\).
Working with logarithmic equations enhances algebraic manipulation skills and is crucial for solving higher-level math problems.

precalculus

Precalculus combines the concepts of algebra and trigonometry to prepare students for the challenges of calculus. It covers topics such as functions, complex numbers, polynomials, and logarithms. A strong grasp of precalculus is essential for students who plan to advance to calculus and beyond.
In the context of our exercise, converting exponential equations to logarithmic form is a vital skill. Understanding the relationship between exponentials and logarithms forms a foundation that will be built upon in calculus when dealing with derivatives and integrals of exponential functions.
Let's recap a part of our exercise: Converting \(10^{-5} = 0.00001\) to logarithmic form gives us \(\log_{10} 0.00001 = -5\). Practicing these conversions strengthens core mathematical understanding and prepares students for more complex topics.

• Developing a strong base in these areas is crucial for mathematical success.
• Utilize logarithms and exponentials for solving real-world problems.