Learn the basics of Logarithm Properties

Logarithm properties are the set of rules that govern how logarithms can be manipulated. These properties are essential for simplifying Logarithmic equations and solving Logarithmic problems, especially those found in ACT prep.

In this short guide, we’ll go over the four most important Logarithm Properties: Product Property, Quotient Property, Power Property, and Logarithm of a Logarithm Property. Each property incldes an example for the understanding. Afterward, we’ll do some practice problems to help you better understand how to use these properties.

 Logrithms are great way to solve the complex problems involving exponential components.

Product Property: The product property states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. It means that instead of multiplication of the components, the logarithm product equal the summation of the logarithmic value of each components which are part of logarithmic multiplication calculation.

 For Example:

logbase(xy)=logbase(x) + logbase (y)

Quotient Property: The quotient property states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator. Similar to the product rule, instead of doing the actual quotient between the component terms, the difference of logarithmic values of the components gives the solution to the equation.

                                    logbase(x/y)=logbase(x) – logbase (y)

Power Property: The next important property in basic properties of logarithmic properties is the power property. The power property states that the logarithm of a power is equal to the product of the logarithm of the base and the exponent. The means the power component of a logarithmic equation becomes the multiplication factor of logarithmic term.

                                    logbase (x2) = 2logbase(x)

Logarithm of a Logarithm Property: The Logarithm of a Logarithm property states that the logarithm of a logarithm is equal to the Logarithm of the Logarithm’s base raised to the power of the Logarithm’s exponent. It means that base of the log is changed and and the previous base becomes the component whose log is the be taken.

logbasex = logax / logabase.

Together, the logarithm properties provide a great set of tools to work around the logarithmic problems. These rules allow to simplify the complex logarithmic problems and then solve.

For instance, take the example below

                                    Logb4.5 = logb(9/2)

                                                = logb9 – logb2

Here first we change the given number into fraction. Then we applied quotient property converting the fraction part of logarithmic equation into a solvable equation.

Remember a few key things while applying the logarithmic properties to complex probems.

  • First check for the quotient rule implications to simplify the equations
  • Look for product rule after the quotient rule
  • Finally solve for the power rule
  • Pay attention to the brackets in moving towards the solution

Also note that logarithmic properties can be applied from left side of equation to right side of equation and vise versa. Whatever can simplify the equation should be taken into the account.

Now that we’ve gone over the four Logarithm Properties, try these practice problems!

1) What is the value of x?


2) Find the value of x for the following equation:



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