Determine If The Function Is Exponential

Determine If The Function Is Exponential

Problems 1-2. Determine if the function is exponential. If so then determine the growth/decay factor. h(x) = x or x^½, or similar functions as indicated.

Problems 3-6. Sketch the graph. Make sure to label at least three points and the horizontal asymptote. The functions include forms like x^n or g(x) = roots or logs as specified.

Problems 7-10. Sketch the graph. Make sure to label at least three points and the vertical asymptote. Functions involve logarithmic forms such as f(x) = log(x) with different bases.

Problems 11-16. Write the logarithm in exponential form. Convert given equations involving logarithms to their exponential equivalents.

Problems 17-22. Write the exponential in logarithmic form. Convert exponential equations to their logarithmic forms.

Problems 23-26. Simplify the given mathematical expressions, often involving nested logarithms and natural logs.

Problems 27-30. Determine the domain and range of each function, expressing answers in interval notation.

Problem 31. Find the amount after 3 years if $1000 is invested at an interest rate of 12% per year, compounded continuously.

Problem 32. A $1200 investment at 7.5% annual interest, compounded monthly for 4 years. Calculate the final account balance.

Problem 33. Expand the logarithm expressions.

Problem 34. Combine multiple logarithmic expressions into a single logarithm.

Paper For Above instruction

Analysis of Exponential Functions and Logarithmic Transformations

The study of exponential and logarithmic functions is fundamental in understanding growth processes, decay, and various real-world phenomena. Determining whether a function is exponential involves examining its form and properties, particularly whether it can be expressed as f(x) = a * b^x, where a is a constant, and b is the base. If so, the function's growth or decay is characterized by the value of b; specifically, if b > 1, the function exhibits exponential growth, whereas if 0

Problem 1 and 2 involve analyzing functions to determine if they are exponential. For example, a function such as h(x) = 2^x clearly is exponential, with base 2, indicating exponential growth, as the output increases rapidly with x. Conversely, h(x) = (1/2)^x would show decay. Recognizing these forms requires inspecting the structure and the coefficients involved. Functions like x^n are polynomial, not exponential, as they involve a power of x rather than an exponential expression.

Sketching the graphs, as in Problems 3-6, involves plotting key points and identifying asymptotes. For exponential functions, the graph typically passes through (0, a) when the function is in the form f(x) = a * b^x, and approaches zero as x tends to negative infinity when 0

Problems 7-10 require sketching graphs of logarithmic functions, which are the inverses of exponential functions. These sketches show the logarithm's growth and decay, with vertical asymptotes at x = 0 for base b > 1, and passing through (1, 0). Proper labeling of points such as (b, 1), (1, 0), and (b^k, k) is necessary to accurately depict behavior. Properly marking asymptotes is essential to understanding the domain restrictions and the nature of the functions.

In Problems 11-16, converting logarithmic equations to exponential form is a vital skill. For instance, log_b(x) = y translates to x = b^y. This conversion helps in solving equations and understanding the inverse relationships. Similarly, natural logs (ln) relate to exponential functions with base e, where ln(x) = y implies x = e^y. Equations such as ln(5) = 8/2 are converted to x = e^(8/2) to facilitate calculations and solve for unknowns.

Problems 17-22 focus on reversing this process: expressing exponential equations in terms of their logarithmic equivalents. For example, e^x = y can be written as ln(y) = x. This allows for solving equations involving exponents using the properties of logarithms, such as product, quotient, and power rules, which simplify complex expressions and solve for unknown variables.

In Problems 23-26, simplifying nested logs involves applying the properties of logarithms to combine or reduce expressions. For example, nested logs like log(log(x)) can be simplified using the rules: log(a * b) = log(a) + log(b), and log(a / b) = log(a) - log(b). When dealing with natural logs or logs of different bases, converting all to a common base facilitates simplification.

Problems 27-30 examine the domain and range of functions involving logs and exponentials. The domain of log_x(x) is restricted to positive real numbers, excluding zero and negatives, with the range typically being all real numbers. When functions involve natural logs or log base b, the domain depends on the argument being positive. The range often includes all real numbers unless restrictions are imposed by the function's form.

Problems 31 and 32 involve compound interest calculations. For continuous compounding, the formula A = P e^(rt) applies, where P is the principal, r the interest rate, and t the time. For monthly compounding, A = P (1 + r/n)^(nt), where n is the number of compounding periods per year. Calculations involve substituting values and solving for the final amount.

Problems 33 and 34 focus on logarithmic expansion and condensation, respectively. Expansion involves applying the laws: ln(ab) = ln(a) + ln(b), ln(a/b) = ln(a) - ln(b), and ln(a^k) = k * ln(a). Condensation involves combining multiple logs into a single logarithm, simplifying complex expressions for easier analysis and calculation.

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