11 Questions
What is the value of x in the equation log3(x) = 2?
x = 9
What is the value of x in the equation log4(x) - log4(2) = 1?
x = 16
Solve for x in the equation log2(x) + log2(3) = 3.
x = 24
Solve for x in the equation 2 * log2(x) = 4.
x = 16
What is the value of x in the equation log2(x) = log3(9)?
x = 8
Solve for x in the equation log3(x) = log2(8).
x = 8
What is the value of x in the equation log2(x) + log2(x) = 5?
x = 32
Solve for x in the equation log3(x) - log3(2) = 1.
x = 9
What is the value of $x$ in the equation $2^x \times 2^{x-1} = 2^5$?
2
What is the value of $x$ in the equation $(3^x)^2 = 3^5$?
5/2
What is the value of $x$ in the equation $\(3^x)^2 = 3^5$?
5/2
Study Notes
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.
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