Please help me solve this equation.
log[2](3)+log[2](x)=log[2](5)+log[2](x-2)
1. Solve equation by setting up algebraically.
2. Use appropriate properties of Logarithm to reduce the number of logarithmic terms to only one.
3. Solve logarithmic equation.
log[2](3)+log[2](x)=log[2](5)+log[2](x-2)
=>
log[2](3*x) = log[2](5*(x-2))
=>
3x=5(x-2)
=>
5x-3x=5*2
=>
2x=10
=>
x=5
Check by putting x into the original equation and see if it balances.
Sure! I can help you solve this equation step by step.
Step 1: Solve the equation by setting up algebraically.
Start by using the properties of logarithms to combine the logarithmic terms with the same base. In this case, we're working with base 2 logarithms. The properties of logarithms allow us to combine logarithms when they have the same base by using addition or subtraction.
So, let's rewrite the equation using these properties:
log[2](3) + log[2](x) = log[2](5) + log[2](x-2)
Applying the properties of logarithms, we can rewrite the left side of the equation as a single logarithm:
log[2](3x) = log[2](5(x-2))
Step 2: Use appropriate properties of logarithms to reduce the number of logarithmic terms to only one.
Since the logarithms on both sides of the equation have the same base (base 2), we can cancel out the logarithms by equating the arguments (the values inside the logarithms).
Therefore, we can write:
3x = 5(x-2)
Step 3: Solve the logarithmic equation.
Next, distribute the 5 on the right side of the equation:
3x = 5x - 10
Now, let's isolate the x term on one side of the equation. Subtract 5x from both sides:
3x - 5x = -10
Simplifying the left side:
-2x = -10
Finally, solve for x by dividing both sides by -2:
x = -10 / -2
x = 5
Therefore, the solution to the equation is x = 5.
To recap:
1. Set up the equation algebraically by combining logarithmic terms with the same base.
2. Use the properties of logarithms to simplify the equation to a single logarithm on each side.
3. Set the arguments of the logarithms equal to each other.
4. Solve the equation to find the value of x.