Logarithmic Functions

Logarithmic Functions Problem Solving


Let w,x,y,zw, x, y, z be positive reals greater than 1, and

logxw=24,logyw=40,logxyzw=12. \log_x w = 24, \log_y w = 40, \log_{xyz} w = 12 .

Evaluate logzw \log_z w .

The only real solution of the equation lnx+ln(x234)=ln72\ln x + \ln (x^2-34)=\ln 72 can be uniquely written in the form a+ba+\sqrt{b} for some integers aa and bb. Find a+ba+b.

Given that log102=0.3010\log_{10}2=0.3010, log103=0.4771\log_{10}3=0.4771 and log107=0.8451\log_{10}7=0.8451, let aa be the number of digits in 3623^{62}. If bb is the first digit fromthe left of 3623^{62}, what is a+ba+b?

Let NN be the number of times that the first digit of 2n2^n is 1,1, with nn being an integer such that 0n10100\leq n \leq 10^{10}. What are the last three digits of N?N?

You may use the fact that
0.3010299956639811952<log102<0.3010299956639811953 0.3010299956639811952 < \log_{10} 2 < 0.3010299956639811953

Evaluate log81+log82+log83++log8154.\lfloor \log_{8} 1 \rfloor + \lfloor \log_{8} 2 \rfloor + \lfloor \log_{8} 3 \rfloor + \cdots + \lfloor \log_{8} 154 \rfloor.

Details and assumptions

Greatest Integer Function / Floor Function: The function x:RZ\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z} refers to the greatest integer smaller than or equal to xx. For example, 2.3=2\lfloor 2.3 \rfloor = 2 and 5=5\lfloor -5 \rfloor = -5.


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