Definition of the Logarithmic Function _-remember your book likes the variable a for base.Characteristics ofMath 112 Steiner sec 4.3 page 1 of 64.3 Logarithmic FunctionsExponential functions and logarithmic functions are inverses of each other. Remember exponential functions are in the formbx= y. If we swap the x’s and y’s we get the functionby=x. This can be re-written in a new form called logarithmic,logbx= y.Definition of the Logarithmic Function _-remember your book likes the variable a for base. Let a be a positive number with a ≠1. The logarithmic function with base a, denoted by logax, is defined by logax= y ↔ ay=xNote: The answer to a logarithm is a power. Calculator functions: log x is the same as log10x with base 10. ln x is the same as lnex with base e. Logarithms and exponential conversions: (logarithms are exponential functions)Exponent Form Logarithmic Form by=x ⇔ logbx= y ey=x ⇔ ln x= y 10y=x ⇔ log x= y (base)power=something⇔ logbase(something)=power Example 1: Write each equation in logarithmic form.a)52=25 b) xy=z c)e7=3 d) ez=kExample 2: Write each equation in exponential form.a)log28=3 b) b=loga√2 c)ln 5=2 d) log x=dMath 112 Steiner sec 4.3 page 2 of 6Remember the answer to a log problem is a power. First we are going to evaluate logs without a calculator.Example 3: Evaluate each without a calculator. Do not give decimal answers. a)log2x= 3 b) log x=2 c)ln x=0 d) log3x=− 2 e) 12=log9x f) logx4=12Properties of General Logarithmic, Common Logarithmic, and Natural Logarithmic General Logarithmic Common Logarithmic (base 10) Natural Logarithmic(base e) 1. logb1=0 1. log 1=0 1. ln 1=0 2. logbb=1 2. log 10 =1 2. ln e=1 3. logbbx=x 3. log 10x=x 3. ln ex=x 4. blogbx=x 4. 10log x= x 4. eln x=xMath 112 Steiner sec 4.3 page 3 of 6Example 4: Evaluate each expression without using a calculator.a)log525=¿ ______ b) ln 1=¿¿____ c)ln e5=¿ _____ RULE: If we have 2 exponential equations equal with the same bases, then their powers must be equal. If bx=by , then x = y. Example 5: Evaluate each expression without using a calculator.a)log5¿ ______ b) ln (1e)=¿ ¿____ c)log8¿ _____ Example 6: Use the calculator to evaluate each of the following. Round to 4 decimal places.a)log 50=¿ b)log 6√2 =c)ln 3=¿¿ d) ln 2+√3=¿¿Math 112 Steiner sec 4.3 page 4 of 6 e) ln ¿ Graphs of Logarithmic Functions: Logarithmic functions and exponential functions are INVERSES.Since the 2 functions are inverses of each other, if we graphed them on the same coordinate system, they would have symmetry of y = x. These 2 functions are graphed below. Graph of exponential and logarithmic functions42-25hx  = xgx  = lnx fx  = ex By looking at the graph, you should notice that the domain of the log functions have been restricted. You cannot take the log of 0 or a negative number. Recall that for y=ax,Domain: (−∞, ∞) Range: (0 , ∞)Intercept: (0 , 1) Horizontal Asymptote: y=0 Characteristics of f(x)=logaxMath 112 Steiner sec 4.3 page 5 of 6Domain: (0 , ∞) Range: (−∞, ∞) x-intercept: (1 , 0)Vertical Asymptote: x = 0 We can still use transformations to graph logs just like we have for other equations.Example 7: Use transformations to graph each. State the domain, range, and asymptote.a)g(x)=2+log5x Domain:Range:Asymptote:b)g(x)=log ¿Domain:Range:Asymptote: IMPORTANT: We have to be very aware of the domain of logarithms. We cannot take the log of 0 or a negative. We will have restricted domains.Example 8: Find the domain of the following functions.a)f(x)=log2x b) f(x)=log2(−x) c) f(x)=log ⁡(x +5)Domain: domain: domain:Math 112 Steiner sec 4.3 page 6 of 6d)f(x)=ln