Solve each equation for x . (a)ln(lnx)=1 (b) e^a x=C e^b x, where a ≠b

Name: Problem 50, Chapter 1: Biocalculus Calculus for the Life Sciences
Uploaded: 2019-06-24T01:02:51-08:00
Duration: 5 min 45 s
Channel: Carlos Pinilla
Description: Solve each equation for x . (a)ln(lnx)=1 (b) e^a x=C e^b x, where a ≠b

step 1 So you want to solve here for a for x on a. So you have the function natural logarithm of natura

Solve each equation for x . (a)ln(lnx)=1 (b) e^a x=C e^b...

Key Concepts

Exponential Equations

Exponential equations are those in which the unknown variable appears in the exponent. Solving these equations often involves taking logarithms to bring down the exponent or using properties of exponents to simplify the equation. Mastery of these techniques is key to isolating and solving for the variable.

Properties of Exponents

Properties of exponents, including the product, quotient, and power rules, are vital for rearranging and simplifying expressions in exponential equations. These rules help in combining or breaking apart exponential terms, which is necessary when solving for variables in equations that compare exponential expressions.

Inverse Functions

Understanding inverse functions is fundamental when solving equations involving logarithms and exponentials. Since the exponential function and the natural logarithm are inverses of each other, applying one after the other cancels the effect of the other. This concept allows us to isolate the variable by 'undoing' the logarithm with an exponential or vice versa.

Natural Logarithm

The natural logarithm, represented by ln, is a logarithm with base e. It plays a critical role in solving equations where the variable is nested within another logarithm or an exponent. Its properties, such as ln(e^x) = x and e^{ln x} = x, are essential tools in rearranging and simplifying such equations.

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