Problem 42 - Intermediate Algebra

Question

Solve each equation. $$ \log 5 x=4 $$

Step-by-Step Solution

Answer

x = 2000

Step 1: Understand the Equation

We are given the equation \( \log 5 x = 4 \). This is a logarithmic equation with the base understood to be 10. Our goal is to solve for \( x \).

Step 2: Convert Logarithmic Equation to Exponential Form

Using the property of logarithms, \( \log_b a = c \) means \( a = b^c \). Here, the equation becomes \( 5x = 10^4 \).

Step 3: Calculate the Exponential Expression

Calculate \( 10^4 \). We know that \( 10^4 = 10000 \). Therefore, the equation becomes \( 5x = 10000 \).

Step 4: Solve for x

Divide both sides of the equation by 5 to isolate \( x \): \( x = \frac{10000}{5} = 2000 \).

Key Concepts

Exponential Form

Understanding exponential form is crucial in solving logarithmic equations. When we talk about exponential form, we are referring to expressing numbers using exponents. In general terms, if we have a function like \( a = b^c \), here \( b \) is the base, \( c \) is the exponent, and \( a \) is the result of the power raised. For instance, if \( b = 10 \) and \( c = 4 \), the exponential form would be \( 10^4 = 10000 \). This is a key step in transforming an equation from logarithmic to exponential form as it allows you to solve for unknown variables within the equation.
When encountering a logarithm, we utilize the reverse process to identify the equivalent exponential expression. In our original example, `\( \log 5x = 4 \)` could be expanded to the expression \( 5x = 10^4 \). This transformation is vital as it simplifies the pathway to finding unknowns, just like the variable \( x \) in our case. It's this conversion to exponential form that makes it possible to directly calculate values that logarithms might otherwise obscure.

Base 10 Logarithms

Base 10 logarithms, denoted \( \log \), are logarithms with a base of 10, unlike natural logarithms which take "e" as their base. Base 10 logarithms are common in solving equations involving powers of 10 or those that require simplification using powers of ten.
Look at this example: \( \log 5x = 4 \). This can easily be confused with having another base, but in mathematics, if no base is specified, it is implicitly base 10.
This means "logarithm of \( 5x \) to the base of 10 equals 4". Base 10 logarithms help us translate exponential forms into simpler numeric forms for calculations. The relationship \( \log_{10} 10000 = 4 \) is the crux that moves the equation solving process forward by converting it into a feasible format that allows algebraic manipulation.
Thus, base 10 logarithms not only offer us a structured way of understanding reactions between exponents and multiplication but also play a significant role in fields like finance and science where calculations with powers of ten are frequent.

Solving Equations

Solving equations, particularly logarithmic ones, involves breaking down the problem into simpler parts. In the equation \( \log 5x = 4 \), our task is to unveil the value of \( x \). We've already translated this to \( 5x = 10^4 \) through exponential form.
Now, solving this involves basic algebraic steps:

• First, compute \( 10^4 = 10000 \).
• Next, resolve \( 5x = 10000 \) by isolating \( x \). This is done by dividing both sides of the equation by 5.
• Through this process, \( x \) equals \( \frac{10000}{5} = 2000 \).

This step-by-step breakdown exemplifies how systematic manipulation of equations, especially logarithmic ones, simplifies the path to a solution. Understanding logarithmic properties and their conversion to exponential equations are essential in solving such mathematical challenges, turning them into comprehendible steps to reach results.