Problem 82 - Algebra 2

Question

Solve each equation. If necessary, round to the nearest ten-thousandth. $$ \frac{1}{2} \log x+\log 4=2 $$

Step-by-Step Solution

Answer

The solution to the equation \(\frac{1}{2} \log x+\log 4=2\) is \(x = 2500\)

Step 1: Combine the Logarithms

The first step is to combine the logarithms using the properties of logarithms. \(\log (ab) = \log(a) + \log(b)\), so we rewrite the equation as: \(\log (4x^{\frac{1}{2}}) = 2\)

Step 2: Change to Exponential Form

We then write the equation in its equivalent exponential form. The base for the log is 10 (since no other base is given), so it becomes: \(10^{2} = 4x^{\frac{1}{2}}\)

Step 3: Solve for x

Next we isolate x by squaring the entire equation which gives: \(x = \frac{10^{4}}{4} = 2500\)

Step 4: Checking the Solution

The last step is to check that the solution is positive since the domain of the logarithm function is \(x > 0\). Since 2500 is a positive number, it is a valid solution.

Key Concepts

Logarithms Properties

Understanding logarithm properties is crucial when solving logarithmic equations. One of the primary properties is the product property:

• \(\log(a) + \log(b) = \log(ab)\)

This property allows you to combine two separate logarithmic terms into one.
If you are given two logs with the same base added together, you can multiply their arguments.
In our example, we have \( \frac{1}{2}\log x + \log 4\). By utilizing the product property, you can rewrite this as \( \log(4 \times x^{\frac{1}{2}})\).
Understanding how to manipulate logarithms with their properties will help you simplify and solve equations effectively.

Exponential Form

Converting a logarithmic equation into its exponential form is another key step in solving these equations. A logarithmic equation \( \log_b(a) = c \) can be rewritten as an exponential equation:

• \( b^c = a \)

Here, \(b\) is the base of the logarithm, which is often 10 in common logarithms unless otherwise specified.
For the problem at hand, after combining the logs, you get \( \log(4x^{\frac{1}{2}}) = 2\). By changing this to its exponential form, you equate it to \(10^2\), giving you \( 10^2 = 4x^{\frac{1}{2}}\).
This translation to exponential form allows you to solve for \(x\) by isolating it on one side of the equation.

Equation Solving

Once you've changed the logarithmic equation into exponential form, the next step is solving for the unknown variable.
In our converted equation from \( 10^2 = 4x^{\frac{1}{2}}\), recognize that you must isolate \(x\) to find its value.
This involves reversing the operations used on \(x\).

• Begin by getting rid of the exponent in \(x^{\frac{1}{2}}\) by squaring both sides of the equation to eliminate the fractional exponent.
• Simplify to get \(x = \frac{10^{4}}{4}\).
• Calculate further to conclude \(x = 2500\).
  

At this point, it is essential to check that your solution makes logical sense within the context, such as ensuring that log arguments are positive.

Rounding Numbers

Rounding numbers, especially to a specific decimal place, is often crucial when dealing with real-world applications of mathematical solutions.
In mathematics, rounding is used to reduce the digits of a number while maintaining its proximity to the original.
This is particularly useful in scenarios where excessive precision is not necessary or when instructions specify rounding requirements.

• The nearest ten-thousandth is four decimal places.
• To round, look at the fifth decimal; if it's 5 or greater, increase the fourth decimal by one.
• Otherwise, keep the fourth decimal as is.
  

Although not needed in our solution since 2500 is an integer, it's a reminder to always check rounding necessities unless specified otherwise.