Problem 12 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-12: Change to logarithmic form. (a) \(10^{4}=10,000\) (b) \(10^{-2}=0.01\) (c) \(10^{x}=38 z\) (d) \(e^{4}=D\) (e) \(e^{0.1 t}=x+2\)

Step-by-Step Solution

Answer

(a) \(\log_{10}{10,000} = 4\), (b) \(\log_{10}{0.01} = -2\), (c) \(\log_{10}{38z} = x\), (d) \(\ln D = 4\), (e) \(\ln(x+2) = 0.1t\).

Step 1: Understanding Logarithmic Form

When changing an expression from exponential form to logarithmic form, you are looking to express the equation in terms of a logarithm. For an expression in the form of \(a^b = c\), the logarithmic form is \( \log_a{c} = b \).

Step 2: Convert (a) to Logarithmic Form

The given exponential form is \(10^{4} = 10,000\). Applying the logarithmic transformation, this becomes \( \log_{10}{10,000} = 4 \).

Step 3: Convert (b) to Logarithmic Form

The given exponential form is \(10^{-2} = 0.01\). Using the conversion, the logarithmic form is \( \log_{10}{0.01} = -2 \).

Step 4: Convert (c) to Logarithmic Form

The given exponential form is \(10^{x} = 38z\). The logarithmic representation is \( \log_{10}{38z} = x \).

Step 5: Convert (d) to Logarithmic Form

The exponential equation given is \(e^{4} = D\). Convert this to logarithmic form using the natural logarithm: \( \ln{D} = 4 \).

Step 6: Convert (e) to Logarithmic Form

For the expression \(e^{0.1 t} = x+2\), the logarithmic form using the natural logarithm is \( \ln{(x+2)} = 0.1t \).

Key Concepts

Understanding Exponential Equations

Exponential equations are equations where variables appear as exponents. These are crucial in describing situations where quantities grow or decay at rates proportional to their current value. For example, growth in population, radioactive decay, or compound interest. An exponential equation is generally expressed as \(a^b = c\), where:

• \(a\) is the base of the exponent,
• \(b\) is the exponent,
• \(c\) is the result.

To solve exponential equations, we often need to transform them into a more solvable form, which is where logarithms come into play. Converting exponential forms to logarithmic forms allows for easier manipulation and understanding of the relationship between the base, exponent, and result.

Natural Logarithms

Natural logarithms are logarithms where the base is the mathematical constant \(e\), approximately equal to 2.718. Denoted as \( \ln \), natural logarithms are widely used in calculus and the natural sciences due to their relationship with rates of change and exponential growth and decay.

• For an expression such as \(e^x = y\), the natural logarithm form is \( \ln{y} = x\).
• The natural logarithm provides a direct way to determine the time constant in growth and decay problems.
• Using \(\ln\), calculations involving exponential functions, especially with base \(e\), become simplified.

Understanding natural logarithms allows us to transition between exponential and logarithmic forms in equations like \(e^{0.1t} = x + 2\), which converts to \(\ln{(x+2)} = 0.1t\), making it more straightforward to solve for \(t\).

Base 10 Logarithms

Base 10 logarithms, or common logarithms, are logarithms with a base of 10. They are frequently used in scientific and engineering fields for simplifying calculations involving large numbers, such as magnitudes of earthquakes or intensity of sounds. Base 10 logarithms are denoted by \( \log \) without a subscript.

• In the expression \(10^b = c\), the base 10 logarithm is \( \log{c} = b\).
• This is particularly useful for representing numbers on a decimal scale, making them easier to interpret and compare.
• In equations like \(10^4 = 10000\), converting to logarithmic form gives us \( \log_{10}{10000} = 4\).

Base 10 logarithms simplify problems involving exponential growth, offering a concise way to handle equations like \(10^{-2} = 0.01\) by transforming it into \(\log_{10}0.01 = -2\). They are essential in any context where powers of 10 are significant.