To solve this equation, we first need to rewrite it using the property of logarithms that allows us to move the coefficients inside the log as exponents.
Therefore, the equation can be rewritten as:
log2(3) + 2log2(x) = log3(81)
Now, we can simplify the equation further by using the properties of logarithms.
First, we know that log2(3) can be rewritten as log3(3)/log3(2) because we can convert between different bases using the change of base formula. This simplifies to log3(3)/log3(2).
Next, since 81 can be expressed as 3^4, we can rewrite log3(81) as 4.
Therefore, our equation now looks like this:
log3(3)/log3(2) + 2log2(x) = 4
Now, we can substitute log2(x) with log3(x)/log3(2) using the change of base formula.
log3(3)/log3(2) + 2(log3(x)/log3(2)) = 4
Now we can combine the terms to simplify the equation:
1 + 2log3(x) - 2 = 4
2log3(x) - 1 = 4
2log3(x) = 5
Now, we can rewrite the equation in exponential form:
3^5 = x^2
243 = x^2
x = ±√243
Therefore, the solutions to the equation are x = ±√243.
To solve this equation, we first need to rewrite it using the property of logarithms that allows us to move the coefficients inside the log as exponents.
Therefore, the equation can be rewritten as:
log2(3) + 2log2(x) = log3(81)
Now, we can simplify the equation further by using the properties of logarithms.
First, we know that log2(3) can be rewritten as log3(3)/log3(2) because we can convert between different bases using the change of base formula. This simplifies to log3(3)/log3(2).
Next, since 81 can be expressed as 3^4, we can rewrite log3(81) as 4.
Therefore, our equation now looks like this:
log3(3)/log3(2) + 2log2(x) = 4
Now, we can substitute log2(x) with log3(x)/log3(2) using the change of base formula.
log3(3)/log3(2) + 2(log3(x)/log3(2)) = 4
Now we can combine the terms to simplify the equation:
1 + 2log3(x) - 2 = 4
2log3(x) - 1 = 4
2log3(x) = 5
Now, we can rewrite the equation in exponential form:
3^5 = x^2
243 = x^2
x = ±√243
Therefore, the solutions to the equation are x = ±√243.