Solving Logarithmic Equations Using Properties of Logarithms Quiz

What is the value of x in the equation log3(x) = 2?

What is the value of x in the equation log4(x) - log4(2) = 1?

Solve for x in the equation log2(x) + log2(3) = 3.

Solve for x in the equation 2 * log2(x) = 4.

What is the value of x in the equation log2(x) = log3(9)?

Solve for x in the equation log3(x) = log2(8).

What is the value of x in the equation log2(x) + log2(x) = 5?

Solve for x in the equation log3(x) - log3(2) = 1.

What is the value of $x$ in the equation $2^x \times 2^{x-1} = 2^5$?

What is the value of $x$ in the equation $(3^x)^2 = 3^5$?

What is the value of $x$ in the equation $\(3^x)^2 = 3^5$?

Solving Logarithmic Equations

Definition

A logarithmic equation is an equation involving logarithms, typically in the form: loga(x) = y
The logarithm is the inverse operation of exponentiation

Properties of Logarithms

loga(M) + loga(N) = loga(MN)
loga(M) - loga(N) = loga(M/N)
k * loga(M) = loga(M^k)

Solving Logarithmic Equations

Method 1: Using Logarithm Properties

Use the properties of logarithms to rewrite the equation in a simpler form
Isolate the logarithm term and apply the inverse operation (exponentiation) to both sides

Example:

log2(x) = 3
2^log2(x) = 2^3
x = 2^3
x = 8

Method 2: Using Exponentiation

Isolate the logarithm term and raise both sides of the equation to the power of the base (a)
Simplify the resulting equation

Example:

log3(x) = 2
3^log3(x) = 3^2
x = 3^2
x = 9

Method 3: Using Logarithmic Identities

Use logarithmic identities, such as loga(x) = logb(x) / logb(a), to rewrite the equation
Simplify the resulting equation

Example:

log2(x) = log4(16)
log2(x) = log2(16) / log2(4)
x = 16

Common Mistakes

Forgetting to isolate the logarithm term before applying exponentiation
Misusing logarithmic properties or identities
Failing to simplify the resulting equation

Test your skills in solving logarithmic equations using different methods, including using logarithm properties, exponentiation, and logarithmic identities. Learn to isolate logarithm terms and apply inverse operations to solve equations.