Answer :
Answer:
7. x = 5
8. x = 1.23963757 or [tex]x=(log_{4}76)-2[/tex]
9. x = 268342.2865 or [tex]x=e^2^5^/^2+5[/tex]
Step-by-step explanation:
Important Info for 7. & 8.
Before we start the problem's it's helpful to remember that logs are the inverse (exact opposite).
Let's say we have an exponential function:
[tex]argument=base^e^x^p^o^n^e^n^t\\y=b^x[/tex],
- where y is the argument (aka answer),
- a is the base,
- and x is the exponent.
When we convert this into a log:
- the base becomes the tiny subscript number next to the log,
- the argument goes beside this,
- and the argument (answer) equals the exponent:
[tex]log_{base}(argument)=exponent[/tex]
[tex]log_{b}(y)=x[/tex]
Important info for 9.
Similar to logs and exponents, the natural log function is the inverse of an exponential function with base e.
Let's say we have an exponential function with base e:
[tex]argument=base^e^x^p^o^n^e^n^t\\y=e^x[/tex]
- where y is the argument (aka answer),
- the base is Euler's number noted by the letter e
- x is the exponent
When we convert this into a log (the specific log for the number e is natural log noted by the letters ln):
- the base (e) becomes the tiny subscript number next to the log,
- the argument goes beside this,
- and the argument (answer) equals the exponent:
[tex]ln_{e}(y)=x[/tex]
7.
Step 1: We need to subtract 2 on both sides to isolate the log:
[tex](2+log_{5}(x)=3)-2\\ log_{5}(x)=1[/tex]
Step 2: Convert it to exponential form, where 5 is the base, 1 is the exponent, and x is the exponent. Then solve for x:
[tex]5^1=x\\5=x[/tex]
Optional Step 3: Check by plugging in 5 for x:
[tex]2+log_{5}(5)=3\\ 2+1=3\\3=3[/tex]
8.
Step 1: We need to add 16 to both sides to isolate the base and exponent:
[tex](4^(^x^+^2^)-16=60)+16\\4^(^x^+^2^)=76[/tex]
Step 2: We need to convert the from exponential to logarithmic form where 4 is the base of the log, 76 is the argument/answer, and (x + 2) is the exponent:
[tex]log_{4}(76)=x+2[/tex]
Step 3: We need to subtract from both sides to solve for x:
[tex](log_{4}(76)=x+2)-2\\log_{4}(76)-2=x\\1.23963757=x[/tex]
Optional Step 4: We can check by plugging in 1.23963757 for x:
[tex]4^(^1^.^2^3^9^6^3^7^5^7^+^2^)-16=60\\76-16=60\\60=60[/tex]
9.
Step 1: We need to divide both sides by 2 to isolate the natural log:
[tex](2ln_{e}(x-5)=25)/2\\ ln_{e}(x-5)=25/2[/tex]
Step 2: We can now convert the natural log to exponential form:
[tex]e^2^5^/^2=x-5[/tex]
Step 3: We must add both 5 to both sides to solve for x:
[tex]e^2^5^/^2+5=x\\268339.2865=x[/tex]
Optional Step 4: We can check by plugging in 268339.2865 for x
[tex]2ln_{e}(268339.2865)=25\\ 2*12.5=25\\25=25[/tex]
Note that [tex]log_{4}(76)-2=x[/tex] (answer for 8.) and e^2^5^/^2+5=x (9.) are more exact answers, whereas 1.239... = x and 268339.2865 = x are approximations, but both the exact and approximate answers were correct when I plugged them in on my graphic calculator.