5.5 Bases Other Than e and Applications

ax = e (ln a)x Definition of Exponential Function to Base a If a is a positive real number (a ≠ 1) and x is any real number, then the exponential function to the base a is denoted by ax and is defined by ax = e (ln a)x If a = 1 , then y = 1x = 1 is a constant function

These functions obey the usual laws of exponents: 2. axay = ax + y 3. ax = ax - y ay 4. (ax)y = axy

logax = ln x ln a Definition of Logarithmic Function to Base a Logarithmic functions to bases other than e can be defined in much the same way as exponential functions to other bases are defined. Definition of Logarithmic Function to Base a If a is a positive real number (a ≠ 1) and x is any positive real number, then the logarithmic function to the base a is denoted by logax and is defined as logax = ln x ln a

loga 1 = 0 loga xy = loga x + loga y loga xn = n loga x Log functions to the base a have properties similar to those of the natural log function: loga 1 = 0 loga xy = loga x + loga y loga xn = n loga x loga x = loga x – loga y y

f(x) = ax and g(x) = loga x are inverse functions Properties of Inverse Functions y = ax if and only if x = loga y alogax = x, for x > 0 loga ax = x, for all x

Examples: Solve for x. 3x = 1 81 2. log2 x = -4 Answer x = -4 x = 1/16

Derivatives for Bases Other than e d [ax ] = (ln a)ax dx 2. d [au ] = (ln a)au du dx dx 3. d [loga x ] = 1 dx (ln a)x 4. d [loga u ] = 1 du dx (ln a)u dx

Answer Examples: Find the derivative of each function: y = 2x y = 23x y = log10 cosx Answer y’ = (ln 2)2x y’ = (3ln 2)23x Answer y’ = -1 tanx ln 10 Answer

Integration of an Exponential Function to a Base Other than e ∫ ax dx = 1 ax + C ln a Find: ∫ 2x dx = 1 2x + C ln 2 Answer

Review of The Power Rule for Real Exponents: Let n be any real number and let u be a differentiable function of x. d [xn] = nxn – 1 dx 2. d [un] = nun – 1 du dx dx

y = ee y = ex y = xe y = xx Find the derivative of each below: Answer y’ = exe - 1 Answer y’ = xx(1 + ln x)