Problem 20 - College Algebra

Question

For the following exercises, use logarithms to solve. $$ 2^{x+1}=5^{2 x-1} $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = \frac{-\ln(2) - \ln(5)}{\ln(2) - 2\ln(5)}\).

Step 1: Apply Logarithms

To solve the equation with exponents, we first apply the logarithm to both sides. This helps in bringing down the exponents where we can manipulate them. So, we take the natural logarithm (ln) for both sides: \[\ln(2^{x+1}) = \ln(5^{2x-1})\]

Step 2: Use Logarithmic Rules

We use the logarithmic identity \(\ln(a^b) = b\ln(a)\) to bring down the exponents in each of the terms:\[(x+1) \ln(2) = (2x-1) \ln(5)\]

Step 3: Expand the Expression

Distribute the logarithms across the terms in the parentheses:\[x \ln(2) + \ln(2) = 2x \ln(5) - \ln(5)\]

Step 4: Rearrange Terms

Rearrange terms to get all \(x\)-related terms on one side and constants on the other:\[x \ln(2) - 2x \ln(5) = -\ln(2) - \ln(5)\]

Step 5: Factor the x Term

Factor out \(x\) from the left side of the equation:\[x(\ln(2) - 2\ln(5)) = -\ln(2) - \ln(5)\]

Step 6: Solve for x

To solve for \(x\), divide both sides by \((\ln(2) - 2\ln(5))\):\[x = \frac{-\ln(2) - \ln(5)}{\ln(2) - 2\ln(5)}\]

Step 7: Simplify the Expression

Calculate the natural logs and simplify the expression numerically, if required, using a calculator.

Key Concepts

Properties of Logarithms

Logarithms play a crucial role in solving exponential equations. One of the key properties of logarithms is that they allow us to transform multiplicative relationships into additive ones. This is immensely helpful, particularly when dealing with complex exponential equations.
Logarithmic properties you should know include:

• Product Rule: \ \log(ab) = \log(a) + \log(b) \
• Quotient Rule: \ \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \
• Power Rule: \ \log(a^b) = b \log(a) \

Understanding these properties is essential for manipulating logarithmic expressions. In our problem, the use of the power rule helps us transform the terms like \(x+1\) \ln(2) \ to a manageable linear equation.
By applying these properties, complex expressions can be simplified, making it easier to solve for the unknown variable.

Exponential Equations

Exponential equations are mathematical expressions where variables appear as exponents. These types of equations often pose a challenge due to their rapid growth rates when variables change. Solving such equations frequently requires the use of logarithms.
In the given problem, the equation \ 2^{x+1}=5^{2x-1} \ represents an exponential equation where the variable \( x \) is in the exponent. To handle this, we take the logarithm of both sides, which helps in making the equation linear.
Here’s a summarized strategy:

• Take the logarithm of both sides. Natural logarithm \( (\ln) \) is often chosen for simplicity.
• Apply logarithmic properties to bring down the exponents, transforming the equation.
• Solve for the variable as you would in a standard linear equation.

Taking logarithms is a powerful way to tackle these problems as it transforms the equation into a more straightforward, algebraic form that can be solved with regular arithmetic techniques.

Natural Logarithm (ln)

The natural logarithm, written as \( \ln \), is a particular type of logarithm that uses the mathematical constant \( e \) (approximately 2.718) as its base. \( \ln \) is commonly used due to its elegant properties and simplicity in calculus and algebra.In our exercise, we applied \( \ln \) to both sides of the equation \( 2^{x+1}=5^{2x-1} \). This choice makes sense because natural logarithms can convert exponential forms into polynomial ones using the power rule: \( \ln(a^b) = b \ln(a) \).
Benefits of using \( \ln \):

• Simplifies differentiation and integration in calculus problems.
• Easily converts exponential terms to linear by using the power rule.
• Widely supported in scientific calculators increasing computational efficiency.

Understanding \( \ln \) is key to solving both theoretical and practical problems in various scientific fields. It simplifies not just solving equations but also understanding the natural growth processes modeled by exponential functions.