Problem 18 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 17-18: Find the number. (a) \(10^{\log 7}\) (b) \(\log 10^{-6}\) (c) \(\log 100,000\) (d) \(\log 0.001\) (e) \(e^{\ln 8}\) (f) \(\ln e^{2 / 3}\) (g) \(e^{1+\ln 5}\)

Step-by-Step Solution

Answer

(a) 7, (b) -6, (c) 5, (d) -3, (e) 8, (f) 2/3, (g) 5e.

Step 1: Simplifying Exponent with Logarithm

For part (a), we have the expression \(10^{\log 7}\). According to the property of logarithms, \(a^{\log_a b} = b\). So, \(10^{\log 7} = 7\).

Step 2: Evaluating Logarithm of a Fraction

For part (b), we have \(\log 10^{-6}\). By the property of logarithms, we have \(\log 10^{n} = n\). Thus, \(\log 10^{-6} = -6\).

Step 3: Calculating Logarithm of Powers of 10

For part (c), \(\log 100,000 \). Since \(100,000 = 10^5\), it follows that \(\log 100,000 = 5\).

Step 4: Evaluating Logarithm with Negative Exponents

For part (d), \(\log 0.001\). We rewrite \(0.001\) as \(10^{-3}\), so \(\log 0.001 = \log 10^{-3} = -3\).

Step 5: Simplifying Exponential Logarithmic Expression

For part (e), \(e^{\ln 8}\). By the property \(a^{\ln_a b} = b\), we have \(e^{\ln 8} = 8\).

Step 6: Simplifying Exponential and Logarithmic Expression

For part (f), \(\ln e^{2/3}\). Using the logarithm property \(\ln e^x = x\), we have \(\ln e^{2/3} = 2/3\).

Step 7: Using Properties of Exponent and Logarithm Combined

For part (g), \(e^{1+\ln 5}\). We use the property \(e^{a+b} = e^a \cdot e^b\). Thus, \(e^{1+\ln 5} = e^1 \cdot e^{\ln 5} = e \cdot 5 = 5e\).

Key Concepts

Logarithmic Properties

Logarithmic properties are rules that simplify the calculation and understanding of logarithms. They are essential in manipulating and solving logarithmic expressions. The properties we often use include:

• Log of a Power: For any number \( x \) and a positive base, \( \log_b(b^x) = x \). This is because the logarithm tells us the power \( b \) must be raised to get \( b^x \). For example, \( \log_{10}(10^5) = 5 \).
• Change of Base: Allows conversion between different logarithmic bases, commonly used to convert to natural logs or common logs.
• Product, Quotient, and Power Rules: These rules break down complex logarithms into manageable parts.

These rules make it easier to evaluate expressions like \( 10^{\log 7} \), simplifying it directly to \( 7 \) due to the property \( a^{\log_a b} = b \). Understanding these properties helps uncloud the mystery behind logarithmic calculations.

Exponents

Exponents are an integral part of understanding logarithms. They serve as the inverse operation to logarithms, working hand-in-hand to solve equations. An exponent denotes the number of times a base is multiplied by itself. For instance, in the expression \( 10^5 \), \( 5 \) is the exponent, meaning \( 10 \) is multiplied by itself five times.
Exponents have various properties:

• Multiplication: \( a^m \cdot a^n = a^{m+n} \), which allows combining exponential terms with the same base.
• Division: \( \frac{a^m}{a^n} = a^{m-n} \).
• Power of a Power: \( (a^m)^n = a^{m \cdot n} \).
• Zero Exponent: \( a^0 = 1 \), for any \( a eq 0 \).

In the context of logarithms, we often use exponents to simplify expressions, such as turning \( e^{1+\ln 5} \) into \( e \cdot e^{\ln 5} \), which equals \( e \cdot 5 = 5e \). This relationship emphasizes the harmony between logarithms and exponents.

Natural Logarithms

Natural logarithms, represented as \( \ln \), are logarithms with the base \( e \), where \( e \approx 2.71828 \). The use of \( e \) is prevalent in continuous growth models, making the natural logarithm highly significant in mathematics and science.
Natural logarithms have unique properties:

• Base of \( e \): \( \ln e = 1 \) because \( e^1 = e \).
• Inverse Property: \( e^{\ln x} = x \), as seen when simplifying expressions like \( e^{\ln 8} \) to \( 8 \).
• Logarithm of \( e^x \): \( \ln e^x = x \), facilitating the simplification of terms such as \( \ln e^{2/3} \) to \( 2/3 \).

These make natural logarithms particularly straightforward when paired with e, simplifying many complex exponential and logarithmic problems.

Logarithmic Expressions

Logarithmic expressions often involve simplifying complex terms using logarithmic properties. A logarithmic expression comprises variables and constants combined through logarithmic operations.
Consider these expressions:

• Positive Powers of 10: \( \log 100,000 = 5 \) because \( 100,000 = 10^5 \).
• Negative Exponents: \( \log 10^{-6} = -6 \), which is intuitive since \( 10^{-6} = \frac{1}{10^6} \).
• Exact Values: Expressions such as \( e^{1+\ln 5} \), which can be simplified using combined logarithmic and exponential properties.

Understanding these expressions equips students with tools to tackle real-world problems, enhancing their mathematical literacy and ability to manipulate complex logarithmic equations effectively.