Express the equation in exponential form

Express the equation in exponential form.

(a) ????(7)=2??

(b) ????(??+6)=-1

The Correct Answer and Explanation is :

Conversion to Exponential Form

To express a logarithmic equation in exponential form, we use the general relationship: log⁡b(a)=cis equivalent tobc=a\log_b(a) = c \quad \text{is equivalent to} \quad b^c = a

Here, bb is the base, aa is the result, and cc is the exponent.

Problem (a)

The equation is log⁡(7)=2x\log(7) = 2x.

• The base of the logarithm is 10 (implied when no base is specified).
• Using the definition of a logarithm, the exponential form is: 102x=710^{2x} = 7

Problem (b)

The equation is log⁡(x+6)=−1\log(x+6) = -1.

• Again, the base of the logarithm is 10 (implied).
• Using the definition of a logarithm, the exponential form is: 10−1=x+610^{-1} = x + 6
• Simplify 10−110^{-1} to 0.10.1, so: x+6=0.1x + 6 = 0.1
• Solving for xx: x=0.1−6=−5.9x = 0.1 – 6 = -5.9

Explanation (300 words)

Logarithmic and exponential functions are inverses of each other. Understanding this relationship allows us to convert between the two forms. In general, if log⁡b(a)=c\log_b(a) = c, it means that bc=ab^c = a. This conversion is essential for solving equations involving logarithms.

For part (a), the logarithmic equation log⁡(7)=2x\log(7) = 2x uses base 10. Applying the definition, the exponential form becomes 102x=710^{2x} = 7. This form expresses the relationship in terms of exponents, which can be useful for solving equations where the variable is in the exponent.

For part (b), log⁡(x+6)=−1\log(x+6) = -1 also uses base 10. Using the same conversion rule, the exponential form is 10−1=x+610^{-1} = x + 6. By simplifying 10−110^{-1} to 0.10.1, we rewrite the equation as x+6=0.1x + 6 = 0.1. Subtracting 6 from both sides gives x=−5.9x = -5.9.

This process highlights the utility of switching between forms. Logarithmic equations are helpful for handling situations where the variable is part of an exponent, while exponential equations can simplify solving for the variable in other contexts. Understanding these forms deepens comprehension of growth, decay, and scaling in mathematical, scientific, and real-world applications.