Skip to main content

Ch. 7 - Transcendental Functions

Hass - Thomas' Calculus 15th Edition

Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Hass 15th Edition

Ch. 7 - Transcendental Functions

문제 7.2.2d

7장, 문제 7.2.2d

2. Express the following logarithms in terms of ln 5 and ln 7.
d. ln 1225

검증된 단계별 안내

1

Start by factoring the number inside the logarithm to express it in terms of its prime factors or known bases. For 1225, find its prime factorization.

Recognize that 1225 can be written as a product of powers of 5 and 7, since 1225 = 35^2 and 35 = 5 \(\times\) 7.

Rewrite the logarithm using the property of logarithms that states \( \ln(a^b) = b \ln(a) \). So, \( \ln(1225) = \ln(35^2) = 2 \ln(35) \).

Next, use the logarithm product rule \( \ln(ab) = \ln(a) + \ln(b) \) to express \( \ln(35) \) as \( \ln(5) + \ln(7) \).

Combine all the steps to express \( \ln(1225) \) entirely in terms of \( \ln(5) \) and \( \ln(7) \) as \( 2 (\ln(5) + \ln(7)) \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

1m

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Properties of Logarithms

Logarithmic properties such as the product, quotient, and power rules allow us to break down complex logarithms into simpler parts. For example, ln(ab) = ln a + ln b and ln(a^n) = n ln a. These rules help express logarithms in terms of known values.

추천 영상:

Change of Base Property

Prime Factorization

Prime factorization involves expressing a number as a product of prime numbers. This is useful in logarithms to rewrite the argument as a product of primes or their powers, enabling the use of logarithmic properties to simplify expressions.

추천 영상:

가이드 코스

Partial Fraction Decomposition: Repeated Linear Factors

Natural Logarithm (ln)

The natural logarithm, denoted ln, is the logarithm with base e. Understanding that ln is a specific logarithm function helps in manipulating and expressing logarithms in terms of known ln values, such as ln 5 and ln 7.

추천 영상:

Derivative of the Natural Logarithmic Function

관련 실천

교과서 질문

2. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?

e. e^(-x)

교과서 질문

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

d. Find the equation for the tangent line to g at the point (f(x_0), x_0) located symmetrically across the 45° line y=x (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line.

68. y= (3x+2)/(2x-11), -2 ≤ x ≤ 2, x_0=1/2

교과서 질문

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

e. Plot the functions f and g, the identity, the two tangent lines, and the line segment joining the points (x_0, f(x_0)) and (f(x_0), x_0). Discuss the symmetries you see across the main diagonal y=x.

72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2

교과서 질문

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

d. Find the equation for the tangent line to g at the point (f(x_0), x_0) located symmetrically across the 45° line y=x (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line.

67. y= √(3x-2), 2/3 ≤ x ≤ 4, x_0=3

교과서 질문

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

d. Find the equation for the tangent line to g at the point (f(x_0), x_0) located symmetrically across the 45° line y=x (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line.

70. y= x³/(x²+1), -1 ≤ x ≤ 1, x_0=1/2

교과서 질문

6. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?

e. x - 2ln(x)