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Equation
372 algebraic function, transcendental function An algebraic function is any function that can be expressed as the sum, difference, product, quotient, power, or root of polynominals. A transcendental function is any function that goes beyond the limits of, or transcends, an algebraic function. 373 properties of real exponents If the bases $a$ and $b$ are positive real numbers and the exponents $x$ and $y$ represent any real numbers, then
  1. $$b^0=1$$
  2. $$b^{-x}=\frac{1}{b^x}$$
  3. $$b^xb^y=b^{x+y}$$
  4. $$(b^x)^y=b^{xy}$$
  5. $$\frac{b^x}{b^y}=b^{x-y}$$
  6. $$(ab)^x=a^xb^x$$
  7. $$\left(\frac{a}{b}\right)^x=\frac{a^x}{b^x}$$
  8. $$\frac{a^{-x}}{b^{-y}}=\frac{b^y}{a^x}$$
373 exponential function If $b$ is a real number and $0<b\ne1,$ (alternatively, $b\in(0,1)\cup(1,-\infty)$) then the function $f$ defined by \[ f(x)=b^x \] is called an exponential function with base $b$. The domain of $f$ is $(-\infty,\infty)$ and the range is $(0,\infty).$ 375 characteristics of the exponential graph
  1. The $y$-intercept is $1$, and the graph has no $x$-intercept.
  2. The $x$-axis is a horizontal asymptote.
  3. If $b>1$, the graph of $f(x)=b^x$ is always increasing.
  4. If $0<b<1$, the graph of $f(x)=b^x$ is always decreasing.
377 exponential function $f$ with base $e.$ \[ e = 2.71828\ldots \] \[ f(x) = e^x \] 378 compound interest Is the interest paid on both principal and any other interest earned previously. Principal $P$, Rate $r$, and where interest is compounded $n$ times per year \[ i = Prt = Pr\frac{1}{n}. \] 379 compound interest formula for $n$ compounding per year If a certain principal $P$ is deposited in a saving account at an interest rate $r$ per year and interest is compounded $n$ times per year, then the amount $A$ in the account after $t$ years is given by the formula \[ A = P\left( 1 + \frac {r}{n}\right)^{nt} \] 380 compound interest formula for continuous compounding If a certain principal $P$ is deposited in a savings account at an interest rate $r$ per year and interest is compounded continously, then the amount $A$ in the account after $t$ years is given by the formula \[ A = Pe^{rt}. \] 383 logarithmic function Let $0 < b \ne 1$ and $x>0.$ The logarithmic function with base $b$ is defined by \[ y=\log_b x \] if and only if \[ b^y=x. \] 384 common logarithm, natural logarithm, Napierian logarithm
  • Common logarithm is the base-10 logarithm ($\log_{10}$) written simply $\log x,\ (x>0).$
  • natural logarithm or Napierian logarithm (named after Scottish mathematician John Napier) is the base-$e$ logarithm ($\log_e$) written as $\ln x$ $(x>0).$
384 logarithms are exponents \[ \log x \text{ means } \log_{10} x \] \[ \ln x \text{ means } \log_e x \] 386 logarithmic identities
  1. $$\log_b 1=0$$
  2. $$\log_b b=1$$
  3. $$\log_b b^x=x,\ (\forall x\in\mathbb{R})$$
  4. $$b^{\log_b x}=x,\ (\forall x\gt0)$$
Note that $x\gt0$ iff $x\in\text{dom}(\log_b)$ 387 logarithmic and exponential forms
  • logarithmic form $$\log_b u = v$$
  • exponential form $$b^v = u$$
Both forms are equivalent. 390 exponential growth, exponential decay
  • Exponential growth formula $$A(t) = A_0 e^{kt}$$
  • Exponential decay formula $$A = Pe^{rt}$$
391 Malthusian model \[ P(t) = P_0 e^{kt} \] 392 half-life The time require for a given mass of a radioactive material to disintegrate to half its original mass. 397 properties of logarithms
  1. $$\log_b xy=\log_b x+\log_b y$$
  2. $$\log_b\frac{x}{y}=\log_b x-\log_b y$$
  3. $$\log_b x^n = n\log_b x$$
399 change of base formula If $\log_b x$ is defined and $0\lt a\neq1,$ then \[ \log_b x = \frac{\log_a x}{\log_a b} \] 400 two identities If we replace $x$ with $a$ in the change of base formula, we obtain \[ \log_b a = \frac {1}{\log_a b} \] or, equivalently, \[ (\log_b a)(\log_a b) = 1 \] 401 logarithmic scales When physical quantities vary over a large range of values, it is convenient to work with logarithmic scales in order to obtain a more manageble set of numbers. 401 magnitude of an earthquake on the Richter scale \[ R = \log \frac {I}{I_0} \] 405 characteristics of the logarithmic graph
  1. The $x$-intercept is $1$, and the graph has no $y$-intercept.
  2. The $y$-axis is a vertical asymptote.
  3. If $b>1$, the graph of $f(x)=\log_b x$ is always increasing.
  4. If $0\lt b\lt1$, the graph of $f(x)=\log_b x$ is always decreasing.
406 graphing related functions Knowing the basic shape of the logarithmic function $f(x) = \log_b x$ enables us to graph several other related functions by applying the shift rules and axis reflection rules. 408 preserve the domain of the original logarithmic function When applying a logarithmic property to a logarithmic function, we must preserve the domain of the original form. To apply a logarithmic property to a function and preserve the same function we must apply the absolute value of $x$. 412 logarithmic equations An equation in which the variable appears in a logarithm. For example, $$\log_3(x-12)=2$$ is equivalent to $$3^2=x-12$$ or $$x=9+12=21.$$ 412 exponential equations An equation in which the variable appears in an exponent. For example, $$e^{x/2}=9$$ is equivalent to $$\ln9=\frac{x}{2}$$ or $$x=2\ln9\approx4.3944.$$ 413 procedure for solving logarithmic equations
  1. Isolate the logarithmic expressions on one side of the equation.
  2. Apply the properties of logarithms, and write the equation in logarithmic form.
  3. Change to exponential form, and solve for the unknown.
  4. Check the solutions. This procedure may produce extraneous roots.
416 procedure for solving exponential equations
  1. Take the common (or natural) logarithm of both sides of the equation.
  2. Apply the properties of logarithms, and write the powers as coefficient of logarithms.
  3. Solve for the unknown, and check the solutions.
417 equations of quadratic type We can solve other logarithmic and exponential equations by recognizing them as an equation of quadractic type. To solve it, follow the steps of rewriting, applying log property #3, factoring, applying zero product property, and replacing.