Write The Exact Answer Using Either Base-10 Or Base-e Logarithms

In various fields, determining the exact answer using either base-10 or base-e logarithms (natural logarithms) is a fundamental skill. Logarithms provide a powerful tool to solve exponential equations, simplify complex calculations, and analyze data in science, engineering, finance, and computer science. Understanding how to manipulate and apply logarithms correctly is essential for obtaining precise results. This article will delve into the detailed methods of writing exact answers using both base-10 and base-e logarithms, providing a comprehensive guide with examples and practical applications.

Understanding Logarithms

Definition of Logarithms

A logarithm is the inverse operation to exponentiation. In simple terms, if b raised to the power of y equals x, then the logarithm base b of x is y. Mathematically, this is expressed as:

b^y = x ⇔ log_b(x) = y

Here:

• b is the base of the logarithm.
• x is the argument of the logarithm.
• y is the exponent or the result of the logarithm.

Common Logarithms (Base-10)

The common logarithm, denoted as log₁₀(x) or simply log(x), has a base of 10. It is widely used in calculations where decimal numbers are prevalent. For example:

log₁₀(100) = 2 because 10² = 100

Natural Logarithms (Base-e)

The natural logarithm, denoted as log_e(x) or ln(x), has a base of e, where e is an irrational number approximately equal to 2.71828. Natural logarithms are extensively used in calculus, physics, and engineering due to their mathematical properties. For example:

ln(e) = 1 because e¹ = e

Properties of Logarithms

Before diving into solving problems, it is crucial to understand the properties of logarithms:

1. Product Rule: log_b(mn) = log_b(m) + log_b(n)
2. Quotient Rule: log_b(m/n) = log_b(m) - log_b(n)
3. Power Rule: log_b(m^p) = p log_b(m)
4. Change of Base Rule: log_b(x) = log_a(x) / log_a(b)

These properties allow us to manipulate logarithmic expressions and simplify calculations, which is vital for obtaining exact answers.

Solving Exponential Equations Using Logarithms

Basic Exponential Equations

Consider the equation:

5^x = 25

To solve for x, we can take the logarithm of both sides. Using base-10 logarithms:

log(5^x) = log(25)

Applying the power rule:

x log(5) = log(25)

x = log(25) / log(5)

Since 25 = 5²:

x = log(5²) / log(5)

x = 2 log(5) / log(5)

x = 2

Alternatively, using natural logarithms:

ln(5^x) = ln(25)

x ln(5) = ln(25)

x = ln(25) / ln(5)

x = ln(5²) / ln(5)

x = 2 ln(5) / ln(5)

x = 2

More Complex Exponential Equations

Consider the equation:

3^{2x - 1} = 81

First, express 81 as a power of 3:

3^{2x - 1} = 3⁴

Since the bases are equal, we can equate the exponents:

2x - 1 = 4

2x = 5

x = 5/2

However, if the right-hand side cannot be easily expressed as a power of the base, logarithms are essential. Consider the equation:

7^x = 20

Taking the common logarithm of both sides:

log(7^x) = log(20)

x log(7) = log(20)

x = log(20) / log(7)

The exact answer is x = log(20) / log(7). This can be approximated using a calculator, but the exact form is retained using logarithms.

Using natural logarithms:

ln(7^x) = ln(20)

x ln(7) = ln(20)

x = ln(20) / ln(7)

The exact answer is x = ln(20) / ln(7).

Exponential Equations with Multiple Terms

Consider the equation:

2 * 3^x = 5^{x + 1}

Taking the natural logarithm of both sides:

ln(2 * 3^x) = ln(5^{x + 1})

Using logarithm properties:

ln(2) + ln(3^x) = (x + 1) ln(5)

ln(2) + x ln(3) = x ln(5) + ln(5)

Rearrange the equation to isolate x:

x ln(3) - x ln(5) = ln(5) - ln(2)

x (ln(3) - ln(5)) = ln(5) - ln(2)

x = (ln(5) - ln(2)) / (ln(3) - ln(5))

x = ln(5/2) / ln(3/5)

The exact answer is x = ln(5/2) / ln(3/5).

Similarly, using common logarithms:

log(2 * 3^x) = log(5^{x + 1})

log(2) + log(3^x) = (x + 1) log(5)

log(2) + x log(3) = x log(5) + log(5)

x (log(3) - log(5)) = log(5) - log(2)

x = (log(5) - log(2)) / (log(3) - log(5))

x = log(5/2) / log(3/5)

The exact answer remains x = log(5/2) / log(3/5).

Applications of Logarithms

Compound Interest

The formula for compound interest is:

A = P (1 + r/n)^nt

Where:

• A is the amount of money accumulated after n years, including interest.
• P is the principal amount (the initial amount of money).
• r is the annual interest rate (as a decimal).
• n is the number of times that interest is compounded per year.
• t is the number of years the money is invested or borrowed for.

Suppose you want to find the time t it takes for an investment to double. Let A = 2P. Then:

2P = P (1 + r/n)^nt

2 = (1 + r/n)^nt

Taking the natural logarithm of both sides:

ln(2) = nt ln(1 + r/n)

t = ln(2) / (n ln(1 + r/n))

This gives the exact expression for the time t.

Radioactive Decay

The formula for radioactive decay is:

N(t) = N₀ *e^-λt

Where:

• N(t) is the amount of substance remaining after time t.
• N₀ is the initial amount of substance.
• λ is the decay constant.
• t is the time.

To find the half-life t_1/2, we set N(t) = N₀/2:

N₀/2 = N₀ *e^-λt_1/2

1/2 = *e^-λt_1/2

Taking the natural logarithm of both sides:

ln(1/2) = -λt_1/2

t_1/2 = ln(1/2) / -λ

t_1/2 = -ln(1/2) / λ

t_1/2 = ln(2) / λ

This gives the exact expression for the half-life.

Decibel Scale

In acoustics and electronics, the decibel (dB) is a logarithmic unit used to express the ratio of two values of a physical quantity, often power or intensity. The formula for decibels is:

dB = 10 log₁₀(P₁/P₀)

Where:

• P₁ is the power being measured.
• P₀ is a reference power.

If you want to find the ratio P₁/P₀ for a given dB value:

dB/10 = log₁₀(P₁/P₀)

P₁/P₀ = 10^dB/10

This provides the exact ratio of powers.

Practical Examples

Example 1: Solving for x in 4^x = 64

Using base-10 logarithms:

log(4^x) = log(64)

x log(4) = log(64)

x = log(64) / log(4)

Since 64 = 4³:

x = log(4³) / log(4)

x = 3 log(4) / log(4)

x = 3

Using natural logarithms:

ln(4^x) = ln(64)

x ln(4) = ln(64)

x = ln(64) / ln(4)

x = ln(4³) / ln(4)

x = 3 ln(4) / ln(4)

x = 3

Example 2: Solving for x in 2^x = 15

Using base-10 logarithms:

log(2^x) = log(15)

x log(2) = log(15)

x = log(15) / log(2)

The exact answer is x = log(15) / log(2).

Using natural logarithms:

ln(2^x) = ln(15)

x ln(2) = ln(15)

x = ln(15) / ln(2)

The exact answer is x = ln(15) / ln(2).

Example 3: Solving for x in 6 e^3x* = 50

First, isolate the exponential term:

e^3x* = 50/6 = 25/3

Taking the natural logarithm of both sides:

ln(e^3x*) = ln(25/3)

3x = ln(25/3)

x = ln(25/3) / 3

The exact answer is x = ln(25/3) / 3.

Example 4: Carbon Dating

Carbon-14 dating is used to determine the age of organic materials. The decay formula is:

N(t) = N₀ *e^-λt

Where λ ≈ 0.000121 per year. Suppose a sample has N(t) = 0.3 N₀. To find the age t:

0. 3 N₀ = N₀ *e^-0.000121t
1. 3 = *e^-0.000121t

Taking the natural logarithm:

ln(0.3) = -0.000121t

t = ln(0.3) / -0.000121

t ≈ 9947 years

The exact answer is t = ln(0.3) / -0.000121 years.

Tips for Working with Logarithms

1. Understand the Properties: Ensure a solid understanding of logarithmic properties to manipulate expressions effectively.
2. Choose the Right Base: Decide whether base-10 or base-e logarithms are more suitable for the problem. Natural logarithms are often preferred in calculus and scientific applications.
3. Isolate Exponential Terms: Before taking logarithms, isolate the exponential term in the equation.
4. Simplify Expressions: Use logarithmic properties to simplify expressions before and after applying logarithms.
5. Keep Exact Answers: Retain the logarithmic form for exact answers and avoid premature approximations.

Common Mistakes to Avoid

1. Incorrectly Applying Logarithmic Properties: Ensure correct application of the product, quotient, and power rules.
2. Forgetting the Base: Always remember the base of the logarithm being used.
3. Mixing Logarithm Bases: Avoid mixing base-10 and base-e logarithms in the same equation without proper conversion.
4. Approximating Too Early: Avoid approximating values before the final step to maintain accuracy.
5. Dividing by Zero: Be cautious of dividing by logarithmic expressions that could be zero.

Conclusion

Mastering the use of logarithms is crucial for solving a wide range of mathematical and scientific problems. This article has provided a comprehensive guide on how to write exact answers using both base-10 and base-e logarithms. By understanding the properties of logarithms, applying them correctly, and avoiding common mistakes, you can confidently tackle complex equations and obtain precise results. Whether you're dealing with exponential equations, compound interest, radioactive decay, or decibel scales, logarithms offer a powerful tool for obtaining exact solutions.