log 1/x + 3log2=log32
adding logs is same as multplying
log((2^3)/x)=log32
take the antilog of each side.
8/x=32
solve for x
log1/x+3log2=log32 log1/x+log2^3=log32 log1/x+log8=log32 according to the law of logarithm +will change to * therefore log(1/x*8)=log32 since the base is equal the they will cancel each other in the equation (1/x*8)=32 (8/x)=32 clear bracket and cross multiply 8=32x divide both sides by 32 to make it x=1/4
To solve the equation log(1/x) + 3log(2) = log(32), we can use the properties of logarithms to simplify and solve for x.
Let's break it down step by step:
Step 1: Apply the logarithmic rule that states log(a * b) = log(a) + log(b). In this case, we have log(32), which can be written as log(2^5) since 32 = 2^5. Therefore, we can rewrite the equation as log(1/x) + 3log(2) = log(2^5).
Step 2: Use the power rule of logarithms, which states log(a^b) = b * log(a). Applying this rule, we can rewrite the equation further as log(1/x) + log(2^3) = log(2^5).
Step 3: Apply the product rule of logarithms, which states log(a) + log(b) = log(a * b). Using this rule, we can rewrite the equation as log((1/x) * 2^3) = log(2^5).
Step 4: Simplify the equation using the logarithmic rules. The equation now becomes log(8/x) = log(32).
Step 5: Since the logarithm of two values is equal, the values inside the logarithm must be equal as well. So we have 8/x = 32.
Step 6: Solve for x by isolating x. Multiply both sides of the equation by x to get rid of the fraction: 8 = 32x.
Step 7: Divide both sides of the equation by 32 to solve for x: x = 8/32.
Step 8: Simplify the fraction: x = 1/4.
Therefore, the solution to the equation log(1/x) + 3log(2) = log(32) is x = 1/4.