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  1. AP Pre Calculus
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What are the differences between linear and logarithmic scales?

Linear: Units are equally spaced, fixed increment. | Logarithmic: Units represent multiplicative change, power of the base.

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What are the differences between linear and logarithmic scales?

Linear: Units are equally spaced, fixed increment. | Logarithmic: Units represent multiplicative change, power of the base.

Compare and contrast log(x)log(x)log(x) and ln(x)ln(x)ln(x).

log(x)log(x)log(x): Base 10 | ln(x)ln(x)ln(x): Base eee (Euler's number).

What are the differences between exponential and logarithmic functions?

Exponential: y=bxy = b^xy=bx, rapid growth | Logarithmic: y=logb(x)y = log_b(x)y=logb​(x), slower growth, inverse of exponential.

Compare and contrast the graphs of y=logb(x)y = log_b(x)y=logb​(x) and y=bxy = b^xy=bx.

y=logb(x)y = log_b(x)y=logb​(x): Vertical asymptote at x=0x = 0x=0, passes through (1, 0) | y=bxy = b^xy=bx: Horizontal asymptote at y=0y = 0y=0, passes through (0, 1)

Compare and contrast the domains of y=logb(x)y = log_b(x)y=logb​(x) and y=bxy = b^xy=bx.

y=logb(x)y = log_b(x)y=logb​(x): Domain is (0,∞)(0, \infty)(0,∞) | y=bxy = b^xy=bx: Domain is (−∞,∞)(-\infty, \infty)(−∞,∞)

Compare and contrast the ranges of y=logb(x)y = log_b(x)y=logb​(x) and y=bxy = b^xy=bx.

y=logb(x)y = log_b(x)y=logb​(x): Range is (−∞,∞)(-\infty, \infty)(−∞,∞) | y=bxy = b^xy=bx: Range is (0,∞)(0, \infty)(0,∞)

Compare and contrast the behavior of y=logb(x)y = log_b(x)y=logb​(x) and y=bxy = b^xy=bx as xxx approaches infinity.

y=logb(x)y = log_b(x)y=logb​(x): Approaches infinity at a decreasing rate | y=bxy = b^xy=bx: Approaches infinity at an increasing rate

Compare and contrast the derivatives of y=log(x)y = log(x)y=log(x) and y=ln(x)y = ln(x)y=ln(x).

y=log(x)y = log(x)y=log(x): Derivative is 1x⋅ln(10)\frac{1}{x \cdot ln(10)}x⋅ln(10)1​ | y=ln(x)y = ln(x)y=ln(x): Derivative is 1x\frac{1}{x}x1​

Compare and contrast the use of linear and logarithmic scales in data representation.

Linear: Suitable for data with evenly distributed values | Logarithmic: Suitable for data with values spanning several orders of magnitude

Compare and contrast the transformations of y=logb(x)y = log_b(x)y=logb​(x) and y=bxy = b^xy=bx when b>1b > 1b>1 and 0<b<10 < b < 10<b<1.

y=logb(x)y = log_b(x)y=logb​(x): Increasing function when b>1b > 1b>1, decreasing function when 0<b<10 < b < 10<b<1 | y=bxy = b^xy=bx: Increasing function when b>1b > 1b>1, decreasing function when 0<b<10 < b < 10<b<1

Define logarithm.

The exponent to which a base must be raised to produce a given number. If ba=cb^a = cba=c, then logb(c)=alog_b(c) = alogb​(c)=a.

What is the base of a common logarithm?

The base of a common logarithm is 10. It is written as log(c)log(c)log(c).

What is the base of a natural logarithm?

The base of a natural logarithm is eee (Euler's number, ≈ 2.71828). It is written as ln(c)ln(c)ln(c).

What is the argument of a logarithm?

The number you're taking the logarithm of.

What is the base of a logarithm?

The base is the number that is raised to a power to obtain the argument. It must be positive and not equal to 1.

Define logarithmic scale.

A scale in which units represent a multiplicative change of the base, where each unit is a power of the base.

What is the logarithm?

The exponent to which you raise the base to get the argument.

What is Euler's Number?

Euler's Number is the base of the natural logarithm, approximately equal to 2.71828.

What is the inverse function of bx=cb^x=cbx=c?

The inverse function is x=logb(c)x = log_b(c)x=logb​(c)

What is the argument of logb(c)log_b(c)logb​(c)?

The argument of logb(c)log_b(c)logb​(c) is ccc.

How to convert 2x=322^x = 322x=32 to logarithmic form?

Identify the base (2), the exponent (x), and the result (32). Rewrite as log2(32)=xlog_2(32) = xlog2​(32)=x.

How to evaluate log5(25)log_5(25)log5​(25)?

Ask: 'To what power must I raise 5 to get 25?' Since 52=255^2 = 2552=25, log5(25)=2log_5(25) = 2log5​(25)=2.

How to solve for xxx in log3(x)=4log_3(x) = 4log3​(x)=4?

Convert to exponential form: 34=x3^4 = x34=x. Then, x=81x = 81x=81.

How to solve the equation log2(x)=5log_2(x) = 5log2​(x)=5 for xxx?

Convert the logarithmic equation to exponential form: 25=x2^5 = x25=x. Simplify to find x=32x = 32x=32.

How to evaluate log(10000)log(10000)log(10000) without a calculator?

Recognize that the base is 10. Determine the power of 10 that equals 10000: 104=1000010^4 = 10000104=10000. Therefore, log(10000)=4log(10000) = 4log(10000)=4.

How to simplify the expression ln(e7)ln(e^7)ln(e7)?

Use the property that ln(ex)=xln(e^x) = xln(ex)=x. Therefore, ln(e7)=7ln(e^7) = 7ln(e7)=7.

How to solve for xxx in the equation logb(x)=ylog_b(x) = ylogb​(x)=y?

Convert to exponential form: by=xb^y = xby=x.

How to solve for xxx in the equation 2log(x)=62log(x) = 62log(x)=6?

Divide both sides by 2: log(x)=3log(x) = 3log(x)=3. Convert to exponential form: 103=x10^3 = x103=x. Therefore, x=1000x = 1000x=1000.

How to solve for xxx in the equation log2(x+1)=3log_2(x + 1) = 3log2​(x+1)=3?

Convert to exponential form: 23=x+12^3 = x + 123=x+1. Simplify: 8=x+18 = x + 18=x+1. Subtract 1 from both sides: x=7x = 7x=7.

How to solve for xxx in the equation ln(x)=0ln(x) = 0ln(x)=0?

Convert to exponential form: e0=xe^0 = xe0=x. Since any number raised to the power of 0 is 1, x=1x = 1x=1.