Solve for x a) 2 log3 x=log3 32+log3 2 b) (3)/(2) log7 x=log7 125 c) log2 x-log2 3=5 d) log6 x=2

step 1 Okay, for this problem we're solving four log problems. Letter A is 2. Log base 3 of x equals lo

Solve for x a) 2 log3 x=log3 32+log3 2 b) (3)/(2) log7...

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Key Concepts

Conversion Between Logarithmic and Exponential Forms

The conversion between logarithmic and exponential forms is a pivotal step in solving logarithmic equations. An equation of the form log_b(A) = C can be rewritten as b^C = A, which transforms the logarithmic relationship into an exponential one. This conversion facilitates straightforward algebraic manipulation, helping to isolate and determine the value of the variable.

Properties of Logarithms

Logarithms follow essential rules such as the product rule (log_b(A) + log_b(B) = log_b(AB)), the quotient rule (log_b(A) - log_b(B) = log_b(A/B)), and the power rule (k * log_b(A) = log_b(A^k)). These rules allow for the manipulation and simplification of logarithmic expressions by combining or breaking apart logarithmic terms, which is fundamental when solving logarithmic equations.

Solving Logarithmic Equations

Solving logarithmic equations involves applying logarithmic properties to consolidate multiple logarithm terms into a single logarithm. Once simplified, the equation can be rewritten in its equivalent exponential form to isolate and solve for the variable. It also requires considering the domain restrictions of logarithms, ensuring that the arguments of all logarithm expressions remain positive.

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