Problem 36 - A Graphical Approach to College Algebra

Question

Evaluate each expression. Do not use a calculator. $$\log 10^{4.3}$$

Step-by-Step Solution

Answer

The value is 4.3.

Step 1: Understand the Basic Logarithm Rule

Recall the logarithmic identity \( \log_b(b^x) = x \). This identity is key in simplifying expressions where the logarithm's base matches the base of the exponent.

Step 2: Identify the Base

Notice that the base of the logarithm in the problem \( \log 10^{4.3} \) is 10.

Step 3: Apply the Logarithm Rule

Using \( \log_{10}(10^{4.3}) = 4.3 \), simplify the expression. Here, the base of the logarithm (10) matches the base of the exponent (10), allowing for direct application of the rule from Step 1.

Step 4: Write the Final Answer

The expression simplifies to \( 4.3 \).

Key Concepts

Logarithms

Logarithms are a mathematical concept used to express the power to which a base must be raised to achieve a certain number. They are essentially the inverse operation of exponentiation. In simple terms, if you have an expression like \( b^x = y \), then \( \log_b(y) = x \), where \( b \) is the base, \( y \) is the result, and \( x \) is the exponent. Logarithms allow us to switch from multiplication and division to addition and subtraction, making them incredibly useful for simplifying complex equations.

• Basic form: \( \log_b(b^x) = x \) indicates that the logarithm of a base raised to an exponent simplifies directly to the exponent \( x \).
• Common bases include 10 (common logarithm, \( \log \)), 2 (binary logarithm), and \( e \) (natural logarithm, \( \ln \)).

Understanding how to work with these expressions is crucial when solving logarithmic problems, as seen in the example where the base 10 was used.

Exponents

Exponents are numbers that denote repeated multiplication of a base number. For instance, \( a^n \) means that \( a \) is multiplied by itself \( n \) times. Exponents play a critical role in expressing large numbers succinctly and are closely related to logarithms.

• Basic rule: \( a^m \times a^n = a^{m+n} \), which means when multiplying like bases, you add the exponents.
• Another rule: \( (a^m)^n = a^{m\times n} \), stating that a power raised to another power multiplies the exponents.

Exponential equations can often be simplified using logarithms, making it easy to solve problems involving powers, such as converting them into linear equations. This technique is vital for simplifying expressions, as shown in the exercise where \( 10^{4.3} \) is evaluated using logarithmic identities.

Simplification Techniques

Simplification techniques are integral to making mathematical expressions easier to work with. Whether you're dealing with logarithms, exponents, or any other mathematical concepts, applying proper simplification rules can turn a complex problem into a manageable one.

For logarithms and exponents, some common techniques include:

• Using logarithmic identities like \( \log_b(b^x) = x \), which helps to quickly simplify expressions when the base of the logarithm is the same as the base of the power.
• Breaking down expressions by applying the rules of exponents, such as multiplication and division rules, to combine or divide like terms.
• Converting exponential forms into logarithmic forms, and vice versa, where necessary to make complicated equations solvable.

In our exercise, the expression \( \log(10^{4.3}) \) simplifies directly to \( 4.3 \) by applying the base rule \( \log_{10}(10^x) = x \), demonstrating how simplification transforms a seemingly daunting task into a straightforward solution.