/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 35 $$ \text { Solve for } t $$ ... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 35

$$ \text { Solve for } t $$ $$ e^{t}=100 $$

Short Answer

Expert verified
t \approx 4.605

Step by step solution

01

- Understand the equation

Recognize that the given equation is an exponential equation in the form of \(e^{t} = 100\). The goal is to solve for the variable \(t\).
02

- Apply the natural logarithm

To isolate \(t\), apply the natural logarithm (ln) to both sides of the equation. \(\ln(e^{t}) = \ln(100)\).
03

- Simplify using the properties of logarithms

Use the property of logarithms that \(\ln(e^{x}) = x\) to simplify the left side of the equation: \(t = \ln(100)\).
04

- Compute the natural logarithm

Use a calculator to find the value of \(\ln(100)\). The approximate value is \(t \approx 4.605\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Functions
Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They are written in the form of:
  • Base^Exponent
For example, in the equation:
  • e^t = 100
'e' is the base, and 't' is the exponent. Exponential functions grow very quickly as the exponent increases. This property makes them useful in modeling real-world phenomena, such as population growth and radioactive decay.
Natural Logarithms
Natural logarithms are logarithms with base 'e', where 'e' is an irrational constant approximately equal to 2.71828. The natural logarithm of a number is usually denoted as 'ln'. It helps us to solve exponential equations by 'undoing' the exponentiation. For example, to solve:
  • e^t = 100
we apply the natural logarithm (ln) to both sides of the equation:
  • ln(e^t) = ln(100)
Using the property that the natural logarithm of 'e' raised to any power is just that power, we simplify this to:
  • t = ln(100)
Logarithmic Properties
Logarithmic properties greatly simplify working with logarithms and exponents. Here are a few key properties:
  • ln(ab) = ln(a) + ln(b)
  • ln(a^b) = b ln(a)
  • ln(e^x) = x
In our exercise, we used the property
  • ln(e^x) = x
This means that for the equation
  • e^t = 100
, by taking the natural logarithm of both sides, we get:
  • ln(e^t) = ln(100)
which simplifies to:
  • t = ln(100)
This powerful property allows us to isolate and solve for the variable in exponential equations.

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Most popular questions from this chapter

The population of Russia dropped from 150 million in 1995 to 144 million in 2002 . Assume the population is decreasing according to the exponential-decay model. 26 a) Find the value of \(k\), and write the equation. b) Estimate the population of Russia in \(2010 .\) c) When will the population of Russia be 100 million?

Pharmaceutical firms spend an immense amount of money in order to test a new medication. After the drug is approved by the Federal Drug Administration, it still takes time for physicians to fully accept and make use of the medication. The use approaches a limiting value of \(100 \%\), or 1 , after time \(t\), in months. Suppose that for a new cancer medication the percentage \(P\) of physicians using the product after \(t\) months is given by $$ P(t)=100 \%\left(1-e^{-0,4 t}\right) $$ a) What percentage of doctors have accepted the medication after 0 mo? 1 mo? 2 mo? 3 mo? 5 mo? 12 mo? 16 mo? b) Find the rate of change \(P^{\prime}(t) .\) c) Sketch a graph of the function.

Differentiate. $$ y=1-e^{-k x} $$

The cumulated oil distribution of a french fry fried at \(185^{\circ}\) for 1 min is approximated by the function $$ G(x)=\frac{100}{1+43.3 e^{-0.0425 x}} $$ where \(x\) is the distance from the surface of the potato, in \(\mu \mathrm{m}\left(1 \mu \mathrm{m}=10^{-6} \mathrm{~m}\right)\), and \(G(x)\) is the percent of oil the potato absorbed within a distance of \(x\) from the surface. For example, \(G(100)=21.1\) indicates that \(21.1 \%\) of the absorbed oil is in the first \(100 \mu \mathrm{m}\) of the potato. \({ }^{21}\) a) Use the formula to compute the percent of oil within \(100 \mu \mathrm{m}\) from the surface, \(150 \mu \mathrm{m}\) from the surface, and \(300 \mu \mathrm{m}\) from the surface. b) Compute the derivative \(G^{\prime}(x)\). c) Find the point of inflection for the function \(G(x)\) d) Sketch the graph of the function.

Differentiate. $$ y=\log _{5} x $$
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