Solve for x : (a) 4^x+2 5^x+1=32000 (b) e^2 x-5 e^x+6=0 (c) logx 36=2 (d) (1)/(2) log(x^2)=5 log

step 1 For this problem, we're going to be solving for x. We have four. Actually, we'll look at E to th

Solve for x : (a) 4^x+2 5^x+1=32000 (b) e^2 x-5 e^x+6=0 (c)...

Key Concepts

Exponential Equations

This concept involves equations where the variable appears in an exponent. Solving such equations typically requires rewriting all exponential terms so that they share the same base, allowing the exponents to be equated. Recognizing and applying the properties of exponents is essential to isolate the variable and solve the equation.

Properties of Exponents

The properties of exponents, such as multiplying powers with the same base (a^(m+n) = a^m · a^n) and raising a power to a power ((a^m)^n = a^(mn)), are fundamental tools. These properties allow us to rewrite and combine exponential expressions, facilitating the simplification of equations where variables appear in the exponents.

Logarithmic Equations

Logarithmic equations contain the unknown variable inside a logarithm. Solving them often involves using the definition of logarithms to convert the equation into exponential form or applying logarithmic identities to combine or separate log terms. This process enables one to isolate the variable and find the solution.

Properties of Logarithms

Key logarithmic properties include the product rule (log(ab) = log a + log b), quotient rule (log(a/b) = log a - log b), and power rule (log(a^b) = b log a). These rules are crucial for manipulating logarithmic expressions, simplifying complex logarithmic equations, and converting between logarithmic and exponential forms.

Transcript

Please give Ace some feedback