Problem 67 - Algebra 2

Question

Consider the equation \(a^{x}=b\) . a. Solve the equation by using log base 10 . b. Solve the equation by using log base \(a\) . c. Use your results in parts (a) and (b) to justify the Change of Base Formula.

Step-by-Step Solution

Answer

In solving the equation \(a^{x}=b\) with log base 10, \(x = log_{10}(b) / log_{10}(a)\) and with log base \(a\), \(x = log_{a}(b)\). These solutions demonstrate the Change of Base Formula, \(log_{b}(a) = log_{c}(a) / log_{c}(b)\), for any valid base \(c\).

Step 1: Solve using log base 10

To solve the equation \(a^{x}=b\), make use of the fact that if \(a^{x}=b\), then \(log_{10}(a^{x}) = log_{10}(b)\). Use the property of logs to transfer the exponent to a multiplier, \(x \cdot log_{10}(a) = log_{10}(b)\). Finally, solve for \(x\) by dividing both sides of the equation by \(log_{10}(a)\), hence \(x = log_{10}(b) / log_{10}(a)\).

Step 2: Solve using log base \(a\)

Similarly, for solving with base \(a\), remember that if \(a^{x}=b\), then \(log_{a}(a^{x}) = log_{a}(b)\). Using the property that \(log_{a}(a^{x}) = x\), the equation simplifies to \(x = log_{a}(b)\).

Step 3: Justify the Change of Base Formula

The Change of Base Formula is \(log_{b}(a) = log_{c}(a) / log_{c}(b)\) for any valid base \(c\).From Steps 1 and 2, \(x = log_{10}(b) / log_{10}(a) = log_{a}(b)\) which demonstrates the Change of Base Formula. This establishes \(log_{b}(a) = log_{c}(a) / log_{c}(b)\), with \(c = 10\), \(b = a\), and \(a = b\).

Key Concepts

Understanding Logarithms

Logarithms are incredibly useful mathematical tools that help us to solve equations involving exponents. Think of a logarithm as the inverse operation to exponentiation. For example, if you know that \(a^x = b\), then finding \(x\) is the same as asking: "a raised to what power gives me \(b\)?" This is exactly what a logarithm does.
Logarithms come with several bases, with common ones being base 10 (common logarithms), base \(e\) (natural logarithms), and any arbitrary base.
An important property of logarithms is that they allow us to "bring down" exponents, making them easier to work with. If you have \(log_{10}(a^x)\), you can use the property \(log_{c}(a^x) = x \cdot log_{c}(a)\). This process is crucial for solving exponential equations where the variable is in the exponent.

Diving into Exponents

At its core, an exponent indicates how many times a number, known as the base, is multiplied by itself. For example, \(a^x\) means that \(a\) is multiplied by itself \(x\) times. Exponents can simplify multiplication and express very large numbers compactly.
However, when the exponent is unknown, solving for it requires tools like logarithms. This is because exponentiation is not straightforwardly reversible like addition or multiplication.
There are some special properties associated with exponents:

• Product Rule: \(a^m \cdot a^n = a^{m+n}\)
• Quotient Rule: \(a^m / a^n = a^{m-n}\)
• Power Rule: \((a^m)^n = a^{m \cdot n}\)

These properties help us transform equations and enable the application of logarithms to solve for variables efficiently.

Exploring Logarithmic Identities

Logarithmic identities are essential tools that simplify logarithmic expressions and help solve complex equations. One such identity is the Change of Base Formula, which allows you to express a logarithm in terms of another base easily. This is particularly useful when you don't have a calculator that supports logarithms for all bases.
The formula is \(log_b(a) = \frac{log_c(a)}{log_c(b)}\) for any valid base \(c\). By letting \(c\) be a common base like 10, you can calculate logs in any other base using a calculator.
Other important logarithmic identities include:

• Product Identity: \(log_c(ab) = log_c(a) + log_c(b)\)
• Quotient Identity: \(log_c\left(\frac{a}{b}\right) = log_c(a) - log_c(b)\)
• Power Identity: \(log_c(a^n) = n \cdot log_c(a)\)

These identities break down complex logarithmic expressions into simpler, more manageable parts, making them incredibly powerful in mathematical computations.