log(4x+9)=1+ log(x-8) please

kraken_they_fear said:
log(4x+9)=1+ log(x-8)
Click to expand...
Against my own judgement lets say these are base 10 logarithms.
Can you see how to reduce the question to \(\displaystyle \log_{10}\left[\dfrac{4x+9}{x-8}\right]=1~?\)
If \(\displaystyle \log_{10}(N)=1\) then \(\displaystyle N=~?\)
 
log(4x+9) = 1+ log(x-8)

Step 1: First change 1 to base 10 logarithm.

log₁₀(4x+9) = log₁₀10 + log₁₀(x-8)

Step 2: Simplify RHS using log rule log₁₀a + log₁₀b = log₁₀(ab)
e.g log₁₀100 + log₁₀10 = log₁₀(10×100)
log₁₀100 + log₁₀10 = log₁₀1000
log₁₀10² + log₁₀10 = log₁₀10³
2log₁₀10 + log₁₀10 = 3log₁₀10
2 + 1 = 3

log₁₀(4x+9) = log₁₀[10(x -8)]

4x+9 = 10(x -8)
4x + 9 = 10x - 80
6x = 89
x = 89/6
= 14⅚
 
hoosie said:
log(4x+9) = 1+ log(x-8)

Step 1: First change 1 to base 10 logarithm.

log₁₀(4x+9) = log₁₀10 + log₁₀(x-8)

Step 2: Simplify RHS using log rule log₁₀a + log₁₀b = log₁₀(ab)
e.g log₁₀100 + log₁₀10 = log₁₀(10×100)
log₁₀100 + log₁₀10 = log₁₀1000
log₁₀10² + log₁₀10 = log₁₀10³
2log₁₀10 + log₁₀10 = 3log₁₀10
2 + 1 = 3

log₁₀(4x+9) = log₁₀[10(x -8)]

4x+9 = 10(x -8)
4x + 9 = 10x - 80
6x = 89
x = 89/6
= 14⅚
Click to expand...

@ hoodie - Please do not give complete solutions when the user has not shown
any attempts. The user was given prompts.
 
lookagain said:
@ hoodie - Please do not give complete solutions when the user has not shown
any attempts. The user was given prompts.
Click to expand...
hoodie, mine is a different concern form lookagain's. Today's mathematics is different from the sort you have been posting here.
You seem to be some thirty years behind the curve on mathematics education. In the mid 1980's to 2000. There was a great movement to take into account the changes in calculators and webbased services. Logarithms are just one example: lead by Gilliam who was president of the MAA thre logarithm is defined as: If \(\displaystyle x>0\text{ then }\log (x) = \int_1^x {\frac{1}{t}dt} \). To see how this definition has been adopted look HERE
If you( or anyone) has doubt of the power of an origination such as the MAA here is my story. We (division of mathematical sciences) adopted that convention for all our courses even the service courses. Well both the chemistry & physics chairs hit the ceilings. Well as luck would have it we were being reviewed by our regional accrediting service. The university was told that we must conform to national standards. So that was that.
hoodie you may want to review modern norms before posting. I plan to call you out on mistakes.
 
pka said:
hoodie, / . . . /
.
.
.
hoodie you may want to review modern norms before posting. I plan to call you out on mistakes.
Click to expand...

The user is hoosie by the way.

pka said:
To see how this definition has been adopted look HERE
Click to expand...

After clicking that link for the Wolfram Alpha treatment, it either shows the WolphramAlpha page with the equation
at the top and immediately moves away from it, or it never goes to that WolphramAlpha page.
 
Last edited:
I more than happy for you to point out where you think I might have gone wrong but in all fields In means the natural logarithm or logₑ( log base e) so that if eˣ= a then In a = x or logₑa = x.
In high school and engineering log usually means the common log or log₁₀ (log base 10) so that if 10ˣ= a then log₁₀a = x.
However, in higher mathematics the common log isn’t very important so for convenience mathematicians often use the notation log to represent the natural logarithm. Wolfram Alpha treats log the same way as do many programming languages. It is the only logarithm which is important for most theoretical purposes. (When I use Wolfam I use the ln and log₁₀ buttons to represent the two logarithms).

I believed the question I answered was from a high school student who would have written In or logₑif a natural logarithm was intended. They wrote log so I assumed a base 10 logarithm was intended. A response from the student would help to clarify.
 
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